(*^

::[	Information =

	"This is a Mathematica Notebook file.  It contains ASCII text, and can be
	transferred by email, ftp, or other text-file transfer utility.  It should
	be read or edited using a copy of Mathematica or MathReader.  If you 
	received this as email, use your mail application or copy/paste to save 
	everything from the line containing (*^ down to the line containing ^*)
	into a plain text file.  On some systems you may have to give the file a 
	name ending with ".ma" to allow Mathematica to recognize it as a Notebook.
	The line below identifies what version of Mathematica created this file,
	but it can be opened using any other version as well.";

	FrontEndVersion = "NeXT Mathematica Notebook Front End Version 2.2";

	NeXTStandardFontEncoding; 
	
	fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8,  24, "Times"; ;
	fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6,  18, "Times"; ;
	fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6,  14, "Times"; ;
	fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20,  18, "Times"; ;
	fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15,  14, "Times"; ;
	fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12,  12, "Times"; ;
	fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1,  12;
	fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1,  10, "Times"; ;
	fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1,  12, "Courier"; ;
	fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5,  12, "Courier"; ;
	fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1,  12, "Courier"; ;
	fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1,  12, "Courier"; ;
	fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1,  12, "Courier"; ;
	fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1,  12, "Courier"; ;
	fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B65535, L1,  10, "Times"; ;
	fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1,  12, "Times"; ;
	fontset = leftheader,  12;
	fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1,  12, "Times"; ;
	fontset = leftfooter,  12;
	fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1,  12;
	fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1,  12;
	fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1,  12, "Courier"; ;
	fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1,  12;
	fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1,  12;
	fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1,  12;
	fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1,  12;
	fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1,  12;
	paletteColors = 128; automaticGrouping; currentKernel; 
]
:[font = title; inactive; preserveAspect; startGroup]
OILSLICK
:[font = section; inactive; preserveAspect; startGroup]
BRIEF ABSTRACT
:[font = subsection; inactive; preserveAspect; endGroup]
Can we use differenced data taken at unknown starting times to ascertain
the size of a growing oil slick?  Either difference or differential equation models will permit discovery if student are willing to plot and do some differencing.
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
GENERAL INFORMATION
:[font = subsection; inactive; preserveAspect; endGroup]
FileName:  OILSLICK

Full title:  Modeling an Oil Slick Growth

Developer: Brian J. Winkel, Department of Mathematics, 
Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA

Contact:  Brian J. Winkel, Department of Mathematics,  Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA.  Phone:  812-877-8412.  Email:  winkel@rose-hulman.edu.   FAX:  812-877-3198.

Support:  
The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849:  Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. 
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
STATEMENT OF PROBLEM
:[font = subsection; inactive; preserveAspect]
An oil-slick spreads at sea.  From time to time, but irregularly, a helicopter  is dispatched. On each trip, it arrives over the slick, the pilot takes a picture, waits  10 minutes,  takes another,  and heads home.   On each of seven trips the size  (in area) of the slick is measured from both photographs.  The data is offered below. 
:[font = subsection; inactive; dontNoPageBreakBelow; noKeepOnOnePage; preserveAspect; endGroup]
       Area of Slick (square miles)

            Initial          10 Minutes 
      Observation           Later 
	1.047		1.136  
	2.005		2.085  
	3.348		3.415  
	5.719		5.762  
	7.273		7.301  
	8.410		8.426  
	9.117		9.126 

(1) Build a model  for  the growth of the oil-slick.  We need a model we 
      can use to predict the size of this slick beyond the observations and we 
      see this as a part of a larger study of oil slick spread.
(2) Confirm that your model is a reasonable one for predicting the growth     
       of this oil slick.
(3) Determine what your model predicts as the long term growth of this 
      oil slick.
(4) Discuss weaknesses, attention to reality, and assumptions you had to 
       make in your model.
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
KEYWORDS 
:[font = subsection; inactive; noKeepOnOnePage; preserveAspect; endGroup]
Data analysis, linear regression, differences, plotting, differential equation (first order, linear), parameter estimating. 
:[font = section; inactive; Cclosed; noKeepOnOnePage; preserveAspect; startGroup]
TEACHER NOTES
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
ISSUES RELATED TO THE PROBLEM
:[font = subsubsection; inactive; preserveAspect]
An early objective in this problem is to get students to plot data as an early strategy for problem-solving.
:[font = subsubsection; inactive; preserveAspect]
The frustration they have with not being able to plot size vs. time should be remembered.  For when they do discover the differencing, the plotting of differences in the size of the slick vs. the size, and the fitting of a linear relationship between a difference and a function to obtain a differential equation, they will see differencing as a valuable tool.
:[font = subsubsection; inactive; preserveAspect]
This project further reinforces the use of a differential equation model. The model is not given to students.  They have to construct it out of data.

Students in traditional approaches in calculus for example, are not called upon to plot data often, indeed, they may not even possess graph paper for class.  However, graph paper can be supplied or one can go right to a CAS or data plotting routines.

:[font = subsubsection; inactive; preserveAspect]
We give the students graph paper at the start of the class and encourage them to use it.  In fact, often we PREFER to be in an environment without CAS so they "mess with" the data on paper.
:[font = subsubsection; inactive; preserveAspect; endGroup]
But it is here that the students get most frustrated for they wish to plot something they have immediately vs. time, e.g. size vs. time.  But the data does not indicate what the absolute time for each observation is, only relative times, e.g., for unknown times t we know size S(t) and size S(t+10) some 10 minutes later only, but we do not know at what time t this actually took place, nor do we know how long there is between observations.  This is a roadblock for the students.  Incidentally, the students will invariably say that the pilot should be fired as she keeps a lousy log!

It is the concept of differencing, i.e., Delta S = S(t+10) - S(t), and realizing that one can plot   Delta S vs. S or Delta S/Delta t vs. S to get a relationship between Delta S/ Delta t and S and thus obtain a differential equation in S.
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
Prerequisites
:[font = subsubsection; inactive; preserveAspect; endGroup]
Plotting data, differencing data, linear regression, first order difference equations, first order differential equations.
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
Time allotment - time management
:[font = subsubsection; inactive; preserveAspect; endGroup]
Students can get to the appropriate differential equations if you work them in small groups and permit diffusion of results about the room all in one class period and then finish the problem at home.  We have done this 4 or 5 times with first year calculus students and differential equations classes.
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
Expectations 
:[font = subsubsection; inactive; preserveAspect; endGroup]
We expect students to plot data, to notice the power of differencing, and to apply their differential equation skills to the problem.
:[font = subsection; inactive; preserveAspect]
Future payoffs
:[font = subsection; inactive; preserveAspect]
Extensions
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
References and Sources
:[font = subsubsection; inactive; preserveAspect; endGroup; endGroup]
 Source: Calculus with Modeling for Science Majors by P. D. Taylor.  In Applied Mathematics  Notes. September 1975 1(2):   64-74. 
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
POSSIBLE SOLUTION(S)
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
We offer up our first modeling solution which leads to a differential equation whose solution will give the size of the slick as a function of time.
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
We enter the Initial Observation Data.
:[font = input; preserveAspect; endGroup]
IO = {1.047,2.005,3.348,5.719,7.273,8.410,9.117};
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
We enter the ten minute Later Observation Data.
:[font = input; preserveAspect; endGroup]
IL = {1.136,2.085,3.415,5.762,7.301,8.426,9.126};
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
We enter the  Difference between Initial Observation Data and the ten minute later Observation Data.  This is an approximation to the derivative of the function OilSize[t] we hope to find, which is the size of the oil slick at time t.
:[font = input; preserveAspect; startGroup]
ID = Table[IL[[i]] - IO[[i]],{i,1,Length[IO]}]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{0.08899999999999996, 0.08000000000000007, 
 
  0.06700000000000017, 0.04299999999999926, 
 
  0.02800000000000046, 0.01600000000000001, 
 
  0.008999999999998564}
;[o]
{0.089, 0.08, 0.067, 0.043, 0.028, 0.016, 0.009}
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
We get ready to plot the difference vs. IO data.
:[font = input; preserveAspect; endGroup]
data = Table[{IO[[i]],ID[[i]]/10},{i,1, Length[IO]}];
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
We see the difference (i.e. the derivative  ~ (IO(t+10) - IO(t))/10) is a linear function of the initial size IO(t). 
:[font = input; preserveAspect; startGroup]
p1 = ListPlot[data,PlotStyle->{PointSize[.02]},
AxesLabel->{"IO(t)", "(IO(t+10)-IO(t))/10"}]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
-0.099752 0.118015 -0.051503 73.575475 [
[.37231 .08315 -3 -9 ]
[.37231 .08315 3 0 ]
[.60834 .08315 -3 -9 ]
[.60834 .08315 3 0 ]
[.84437 .08315 -3 -9 ]
[.84437 .08315 3 0 ]
[1.025 .09565 0 -4.5 ]
[1.025 .09565 30 4.5 ]
[.12378 .2428 -30 -4.5 ]
[.12378 .2428 0 4.5 ]
[.12378 .38995 -30 -4.5 ]
[.12378 .38995 0 4.5 ]
[.12378 .5371 -30 -4.5 ]
[.12378 .5371 0 4.5 ]
[.13628 .64303 -57 0 ]
[.13628 .64303 57 9 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.37231 .09565 m
.37231 .1019 L
s
[(4)] .37231 .08315 0 1 Mshowa
.60834 .09565 m
.60834 .1019 L
s
[(6)] .60834 .08315 0 1 Mshowa
.84437 .09565 m
.84437 .1019 L
s
[(8)] .84437 .08315 0 1 Mshowa
.125 Mabswid
.19529 .09565 m
.19529 .0994 L
s
.25429 .09565 m
.25429 .0994 L
s
.3133 .09565 m
.3133 .0994 L
s
.43132 .09565 m
.43132 .0994 L
s
.49032 .09565 m
.49032 .0994 L
s
.54933 .09565 m
.54933 .0994 L
s
.66735 .09565 m
.66735 .0994 L
s
.72635 .09565 m
.72635 .0994 L
s
.78536 .09565 m
.78536 .0994 L
s
.07727 .09565 m
.07727 .0994 L
s
.01826 .09565 m
.01826 .0994 L
s
.90338 .09565 m
.90338 .0994 L
s
.96238 .09565 m
.96238 .0994 L
s
.25 Mabswid
0 .09565 m
1 .09565 L
s
[(IO\(t\))] 1.025 .09565 -1 0 Mshowa
.13628 .2428 m
.14253 .2428 L
s
[(0.004)] .12378 .2428 1 0 Mshowa
.13628 .38995 m
.14253 .38995 L
s
[(0.006)] .12378 .38995 1 0 Mshowa
.13628 .5371 m
.14253 .5371 L
s
[(0.008)] .12378 .5371 1 0 Mshowa
.125 Mabswid
.13628 .13244 m
.14003 .13244 L
s
.13628 .16922 m
.14003 .16922 L
s
.13628 .20601 m
.14003 .20601 L
s
.13628 .27959 m
.14003 .27959 L
s
.13628 .31637 m
.14003 .31637 L
s
.13628 .35316 m
.14003 .35316 L
s
.13628 .42674 m
.14003 .42674 L
s
.13628 .46353 m
.14003 .46353 L
s
.13628 .50031 m
.14003 .50031 L
s
.13628 .05886 m
.14003 .05886 L
s
.13628 .02207 m
.14003 .02207 L
s
.13628 .57389 m
.14003 .57389 L
s
.13628 .61068 m
.14003 .61068 L
s
.25 Mabswid
.13628 0 m
.13628 .61803 L
s
[(\(IO\(t+10\)-IO\(t\)\)/10)] .13628 .64303 0 -1 Mshowa
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.02 w
.02381 .60332 Mdot
.13687 .5371 Mdot
.29536 .44145 Mdot
.57518 .26487 Mdot
.75857 .15451 Mdot
.89275 .06622 Mdot
.97619 .01472 Mdot
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
Graphics["<<>>"]
;[o]
-Graphics-
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
And we determine the linear relationship from a best fit command in Mathematica.
;[s]
3:0,0;68,1;79,2;80,-1;
3:1,10,8,Times,1,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,1,12,0,0,0;
:[font = input; preserveAspect; startGroup]
pfit[x_] = Fit[data,{1,x},x]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
0.009985361121877566 - 0.0009940011336478001*x
;[o]
0.00998536 - 0.000994001 x
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
We plot the line and then the data and the line and see that the fit looks quite good.
:[font = input; preserveAspect; startGroup]
p2 = Plot[pfit[x],{x,0,10}]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.02381 0.095238 0.014715 58.946674 [
[.21429 .00222 -3 -9 ]
[.21429 .00222 3 0 ]
[.40476 .00222 -3 -9 ]
[.40476 .00222 3 0 ]
[.59524 .00222 -3 -9 ]
[.59524 .00222 3 0 ]
[.78571 .00222 -3 -9 ]
[.78571 .00222 3 0 ]
[.97619 .00222 -6 -9 ]
[.97619 .00222 6 0 ]
[.01131 .13261 -30 -4.5 ]
[.01131 .13261 0 4.5 ]
[.01131 .2505 -30 -4.5 ]
[.01131 .2505 0 4.5 ]
[.01131 .3684 -30 -4.5 ]
[.01131 .3684 0 4.5 ]
[.01131 .48629 -30 -4.5 ]
[.01131 .48629 0 4.5 ]
[.01131 .60418 -24 -4.5 ]
[.01131 .60418 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.21429 .01472 m
.21429 .02097 L
s
[(2)] .21429 .00222 0 1 Mshowa
.40476 .01472 m
.40476 .02097 L
s
[(4)] .40476 .00222 0 1 Mshowa
.59524 .01472 m
.59524 .02097 L
s
[(6)] .59524 .00222 0 1 Mshowa
.78571 .01472 m
.78571 .02097 L
s
[(8)] .78571 .00222 0 1 Mshowa
.97619 .01472 m
.97619 .02097 L
s
[(10)] .97619 .00222 0 1 Mshowa
.125 Mabswid
.07143 .01472 m
.07143 .01847 L
s
.11905 .01472 m
.11905 .01847 L
s
.16667 .01472 m
.16667 .01847 L
s
.2619 .01472 m
.2619 .01847 L
s
.30952 .01472 m
.30952 .01847 L
s
.35714 .01472 m
.35714 .01847 L
s
.45238 .01472 m
.45238 .01847 L
s
.5 .01472 m
.5 .01847 L
s
.54762 .01472 m
.54762 .01847 L
s
.64286 .01472 m
.64286 .01847 L
s
.69048 .01472 m
.69048 .01847 L
s
.7381 .01472 m
.7381 .01847 L
s
.83333 .01472 m
.83333 .01847 L
s
.88095 .01472 m
.88095 .01847 L
s
.92857 .01472 m
.92857 .01847 L
s
.25 Mabswid
0 .01472 m
1 .01472 L
s
.02381 .13261 m
.03006 .13261 L
s
[(0.002)] .01131 .13261 1 0 Mshowa
.02381 .2505 m
.03006 .2505 L
s
[(0.004)] .01131 .2505 1 0 Mshowa
.02381 .3684 m
.03006 .3684 L
s
[(0.006)] .01131 .3684 1 0 Mshowa
.02381 .48629 m
.03006 .48629 L
s
[(0.008)] .01131 .48629 1 0 Mshowa
.02381 .60418 m
.03006 .60418 L
s
[(0.01)] .01131 .60418 1 0 Mshowa
.125 Mabswid
.02381 .04419 m
.02756 .04419 L
s
.02381 .07366 m
.02756 .07366 L
s
.02381 .10314 m
.02756 .10314 L
s
.02381 .16208 m
.02756 .16208 L
s
.02381 .19156 m
.02756 .19156 L
s
.02381 .22103 m
.02756 .22103 L
s
.02381 .27998 m
.02756 .27998 L
s
.02381 .30945 m
.02756 .30945 L
s
.02381 .33892 m
.02756 .33892 L
s
.02381 .39787 m
.02756 .39787 L
s
.02381 .42734 m
.02756 .42734 L
s
.02381 .45682 m
.02756 .45682 L
s
.02381 .51576 m
.02756 .51576 L
s
.02381 .54524 m
.02756 .54524 L
s
.02381 .57471 m
.02756 .57471 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.5 Mabswid
.02381 .60332 m
.06244 .57955 L
.10458 .55363 L
.14415 .52928 L
.18221 .50587 L
.22272 .48095 L
.26171 .45696 L
.30316 .43146 L
.34309 .40689 L
.3815 .38326 L
.42237 .35812 L
.46172 .33391 L
.49955 .31063 L
.53984 .28584 L
.57861 .26199 L
.61984 .23663 L
.65954 .2122 L
.69774 .1887 L
.73838 .16369 L
.77751 .13962 L
.81909 .11404 L
.85916 .08939 L
.89771 .06567 L
.93871 .04044 L
.97619 .01739 L
s
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup]
Graphics["<<>>"]
;[o]
-Graphics-
:[font = input; preserveAspect; startGroup]
Show[p1,p2]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.02381 0.095238 0.014715 58.946674 [
[.21429 .00222 -3 -9 ]
[.21429 .00222 3 0 ]
[.40476 .00222 -3 -9 ]
[.40476 .00222 3 0 ]
[.59524 .00222 -3 -9 ]
[.59524 .00222 3 0 ]
[.78571 .00222 -3 -9 ]
[.78571 .00222 3 0 ]
[.97619 .00222 -6 -9 ]
[.97619 .00222 6 0 ]
[1.025 .01472 0 -4.5 ]
[1.025 .01472 30 4.5 ]
[.01131 .13261 -30 -4.5 ]
[.01131 .13261 0 4.5 ]
[.01131 .2505 -30 -4.5 ]
[.01131 .2505 0 4.5 ]
[.01131 .3684 -30 -4.5 ]
[.01131 .3684 0 4.5 ]
[.01131 .48629 -30 -4.5 ]
[.01131 .48629 0 4.5 ]
[.01131 .60418 -24 -4.5 ]
[.01131 .60418 0 4.5 ]
[.02381 .64303 -57 0 ]
[.02381 .64303 57 9 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.21429 .01472 m
.21429 .02097 L
s
[(2)] .21429 .00222 0 1 Mshowa
.40476 .01472 m
.40476 .02097 L
s
[(4)] .40476 .00222 0 1 Mshowa
.59524 .01472 m
.59524 .02097 L
s
[(6)] .59524 .00222 0 1 Mshowa
.78571 .01472 m
.78571 .02097 L
s
[(8)] .78571 .00222 0 1 Mshowa
.97619 .01472 m
.97619 .02097 L
s
[(10)] .97619 .00222 0 1 Mshowa
.125 Mabswid
.07143 .01472 m
.07143 .01847 L
s
.11905 .01472 m
.11905 .01847 L
s
.16667 .01472 m
.16667 .01847 L
s
.2619 .01472 m
.2619 .01847 L
s
.30952 .01472 m
.30952 .01847 L
s
.35714 .01472 m
.35714 .01847 L
s
.45238 .01472 m
.45238 .01847 L
s
.5 .01472 m
.5 .01847 L
s
.54762 .01472 m
.54762 .01847 L
s
.64286 .01472 m
.64286 .01847 L
s
.69048 .01472 m
.69048 .01847 L
s
.7381 .01472 m
.7381 .01847 L
s
.83333 .01472 m
.83333 .01847 L
s
.88095 .01472 m
.88095 .01847 L
s
.92857 .01472 m
.92857 .01847 L
s
.25 Mabswid
0 .01472 m
1 .01472 L
s
[(IO\(t\))] 1.025 .01472 -1 0 Mshowa
.02381 .13261 m
.03006 .13261 L
s
[(0.002)] .01131 .13261 1 0 Mshowa
.02381 .2505 m
.03006 .2505 L
s
[(0.004)] .01131 .2505 1 0 Mshowa
.02381 .3684 m
.03006 .3684 L
s
[(0.006)] .01131 .3684 1 0 Mshowa
.02381 .48629 m
.03006 .48629 L
s
[(0.008)] .01131 .48629 1 0 Mshowa
.02381 .60418 m
.03006 .60418 L
s
[(0.01)] .01131 .60418 1 0 Mshowa
.125 Mabswid
.02381 .04419 m
.02756 .04419 L
s
.02381 .07366 m
.02756 .07366 L
s
.02381 .10314 m
.02756 .10314 L
s
.02381 .16208 m
.02756 .16208 L
s
.02381 .19156 m
.02756 .19156 L
s
.02381 .22103 m
.02756 .22103 L
s
.02381 .27998 m
.02756 .27998 L
s
.02381 .30945 m
.02756 .30945 L
s
.02381 .33892 m
.02756 .33892 L
s
.02381 .39787 m
.02756 .39787 L
s
.02381 .42734 m
.02756 .42734 L
s
.02381 .45682 m
.02756 .45682 L
s
.02381 .51576 m
.02756 .51576 L
s
.02381 .54524 m
.02756 .54524 L
s
.02381 .57471 m
.02756 .57471 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
[(\(IO\(t+10\)-IO\(t\)\)/10)] .02381 .64303 0 -1 Mshowa
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.02 w
.12352 .53934 Mdot
.21476 .48629 Mdot
.34267 .40966 Mdot
.56848 .26819 Mdot
.71648 .17977 Mdot
.82476 .10903 Mdot
.8921 .06777 Mdot
.5 Mabswid
.02381 .60332 m
.06244 .57955 L
.10458 .55363 L
.14415 .52928 L
.18221 .50587 L
.22272 .48095 L
.26171 .45696 L
.30316 .43146 L
.34309 .40689 L
.3815 .38326 L
.42237 .35812 L
.46172 .33391 L
.49955 .31063 L
.53984 .28584 L
.57861 .26199 L
.61984 .23663 L
.65954 .2122 L
.69774 .1887 L
.73838 .16369 L
.77751 .13962 L
.81909 .11404 L
.85916 .08939 L
.89771 .06567 L
.93871 .04044 L
.97619 .01739 L
s
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
Graphics["<<>>"]
;[o]
-Graphics-
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
Thus we have an approximation to a differential equation in the function OilSize[t] which we proceed to solve.
:[font = input; preserveAspect; endGroup]
eqDe = OilSize'[t] == pfit[OilSize[t]];
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
We solve and pick off the solution to the differential equation.
:[font = input; preserveAspect]
sol = DSolve[{eqDe,OilSize[0] == 1.047},OilSize[t],t];
:[font = input; preserveAspect; endGroup]
OilSlick[t_] = OilSize[t]/.sol[[1]];
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
We plot the solution function called OilSlick[t].
:[font = input; preserveAspect; startGroup]
pOil = Plot[OilSlick[t],{t,0,4000}]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.02381 0.000238 0.014715 0.059595 [
[.2619 .00222 -12 -9 ]
[.2619 .00222 12 0 ]
[.5 .00222 -12 -9 ]
[.5 .00222 12 0 ]
[.7381 .00222 -12 -9 ]
[.7381 .00222 12 0 ]
[.97619 .00222 -12 -9 ]
[.97619 .00222 12 0 ]
[.01131 .1339 -6 -4.5 ]
[.01131 .1339 0 4.5 ]
[.01131 .25309 -6 -4.5 ]
[.01131 .25309 0 4.5 ]
[.01131 .37228 -6 -4.5 ]
[.01131 .37228 0 4.5 ]
[.01131 .49147 -6 -4.5 ]
[.01131 .49147 0 4.5 ]
[.01131 .61066 -12 -4.5 ]
[.01131 .61066 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.2619 .01472 m
.2619 .02097 L
s
[(1000)] .2619 .00222 0 1 Mshowa
.5 .01472 m
.5 .02097 L
s
[(2000)] .5 .00222 0 1 Mshowa
.7381 .01472 m
.7381 .02097 L
s
[(3000)] .7381 .00222 0 1 Mshowa
.97619 .01472 m
.97619 .02097 L
s
[(4000)] .97619 .00222 0 1 Mshowa
.125 Mabswid
.07143 .01472 m
.07143 .01847 L
s
.11905 .01472 m
.11905 .01847 L
s
.16667 .01472 m
.16667 .01847 L
s
.21429 .01472 m
.21429 .01847 L
s
.30952 .01472 m
.30952 .01847 L
s
.35714 .01472 m
.35714 .01847 L
s
.40476 .01472 m
.40476 .01847 L
s
.45238 .01472 m
.45238 .01847 L
s
.54762 .01472 m
.54762 .01847 L
s
.59524 .01472 m
.59524 .01847 L
s
.64286 .01472 m
.64286 .01847 L
s
.69048 .01472 m
.69048 .01847 L
s
.78571 .01472 m
.78571 .01847 L
s
.83333 .01472 m
.83333 .01847 L
s
.88095 .01472 m
.88095 .01847 L
s
.92857 .01472 m
.92857 .01847 L
s
.25 Mabswid
0 .01472 m
1 .01472 L
s
.02381 .1339 m
.03006 .1339 L
s
[(2)] .01131 .1339 1 0 Mshowa
.02381 .25309 m
.03006 .25309 L
s
[(4)] .01131 .25309 1 0 Mshowa
.02381 .37228 m
.03006 .37228 L
s
[(6)] .01131 .37228 1 0 Mshowa
.02381 .49147 m
.03006 .49147 L
s
[(8)] .01131 .49147 1 0 Mshowa
.02381 .61066 m
.03006 .61066 L
s
[(10)] .01131 .61066 1 0 Mshowa
.125 Mabswid
.02381 .04451 m
.02756 .04451 L
s
.02381 .07431 m
.02756 .07431 L
s
.02381 .10411 m
.02756 .10411 L
s
.02381 .1637 m
.02756 .1637 L
s
.02381 .1935 m
.02756 .1935 L
s
.02381 .2233 m
.02756 .2233 L
s
.02381 .28289 m
.02756 .28289 L
s
.02381 .31269 m
.02756 .31269 L
s
.02381 .34249 m
.02756 .34249 L
s
.02381 .40208 m
.02756 .40208 L
s
.02381 .43188 m
.02756 .43188 L
s
.02381 .46167 m
.02756 .46167 L
s
.02381 .52127 m
.02756 .52127 L
s
.02381 .55107 m
.02756 .55107 L
s
.02381 .58086 m
.02756 .58086 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.5 Mabswid
.02381 .07711 m
.06244 .15699 L
.10458 .23061 L
.14415 .2889 L
.18221 .33656 L
.22272 .37963 L
.26171 .41475 L
.30316 .44631 L
.34309 .47196 L
.3815 .49292 L
.42237 .51181 L
.46172 .5272 L
.49955 .53979 L
.53984 .55118 L
.57861 .56048 L
.61984 .56884 L
.65954 .57565 L
.69774 .58121 L
.73838 .58623 L
.77751 .59032 L
.81909 .59399 L
.85916 .59698 L
.89771 .59942 L
.93871 .60162 L
.97619 .60332 L
s
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
Graphics["<<>>"]
;[o]
-Graphics-
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
Now we determine the times at which our OilSlick[t] function is at the IO times.
:[font = input; preserveAspect; endGroup]
time = Table[t/.FindRoot[IO[[i]]== OilSlick[t],
	{t,2000}][[1]], {i,1, Length[IO]}];
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
And we list these times, t, and the corresponding OilSlick[t] values.
:[font = input; preserveAspect; startGroup]
newdata = Table[{time[[i]],IO[[i]]},{i,1, Length[IO]}]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{{-5.963157726635026*^-8, 1.046999999999999}, 
 
  {113.2443279192359, 2.004999999999999}, 
 
  {297.1011299327295, 3.347999999999999}, 
 
  {736.7034186550443, 5.719}, 
 
  {1184.382596556388, 7.272999999999999}, 
 
  {1715.337176596304, 8.41}, 
 
  {2284.829713015071, 9.117}}
;[o]
             -8
{{-5.96316 10  , 1.047}, {113.244, 2.005}, 
 
  {297.101, 3.348}, {736.703, 5.719}, 
 
  {1184.38, 7.273}, {1715.34, 8.41}, 
 
  {2284.83, 9.117}}
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
with a plot to follow of the IO data over the function OilSLick[t].
:[font = input; preserveAspect; startGroup]
p1new = ListPlot[newdata,PlotStyle->{PointSize[.02]}]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.02381 0.000417 -0.06165 0.072937 [
[.23222 .07172 -9 -9 ]
[.23222 .07172 9 0 ]
[.44064 .07172 -12 -9 ]
[.44064 .07172 12 0 ]
[.64905 .07172 -12 -9 ]
[.64905 .07172 12 0 ]
[.85747 .07172 -12 -9 ]
[.85747 .07172 12 0 ]
[.01131 .2301 -6 -4.5 ]
[.01131 .2301 0 4.5 ]
[.01131 .37597 -6 -4.5 ]
[.01131 .37597 0 4.5 ]
[.01131 .52185 -6 -4.5 ]
[.01131 .52185 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.23222 .08422 m
.23222 .09047 L
s
[(500)] .23222 .07172 0 1 Mshowa
.44064 .08422 m
.44064 .09047 L
s
[(1000)] .44064 .07172 0 1 Mshowa
.64905 .08422 m
.64905 .09047 L
s
[(1500)] .64905 .07172 0 1 Mshowa
.85747 .08422 m
.85747 .09047 L
s
[(2000)] .85747 .07172 0 1 Mshowa
.125 Mabswid
.06549 .08422 m
.06549 .08797 L
s
.10718 .08422 m
.10718 .08797 L
s
.14886 .08422 m
.14886 .08797 L
s
.19054 .08422 m
.19054 .08797 L
s
.27391 .08422 m
.27391 .08797 L
s
.31559 .08422 m
.31559 .08797 L
s
.35727 .08422 m
.35727 .08797 L
s
.39895 .08422 m
.39895 .08797 L
s
.48232 .08422 m
.48232 .08797 L
s
.524 .08422 m
.524 .08797 L
s
.56569 .08422 m
.56569 .08797 L
s
.60737 .08422 m
.60737 .08797 L
s
.69073 .08422 m
.69073 .08797 L
s
.73242 .08422 m
.73242 .08797 L
s
.7741 .08422 m
.7741 .08797 L
s
.81578 .08422 m
.81578 .08797 L
s
.89915 .08422 m
.89915 .08797 L
s
.94083 .08422 m
.94083 .08797 L
s
.98251 .08422 m
.98251 .08797 L
s
.25 Mabswid
0 .08422 m
1 .08422 L
s
.02381 .2301 m
.03006 .2301 L
s
[(4)] .01131 .2301 1 0 Mshowa
.02381 .37597 m
.03006 .37597 L
s
[(6)] .01131 .37597 1 0 Mshowa
.02381 .52185 m
.03006 .52185 L
s
[(8)] .01131 .52185 1 0 Mshowa
.125 Mabswid
.02381 .12069 m
.02756 .12069 L
s
.02381 .15716 m
.02756 .15716 L
s
.02381 .19363 m
.02756 .19363 L
s
.02381 .26657 m
.02756 .26657 L
s
.02381 .30304 m
.02756 .30304 L
s
.02381 .3395 m
.02756 .3395 L
s
.02381 .41244 m
.02756 .41244 L
s
.02381 .44891 m
.02756 .44891 L
s
.02381 .48538 m
.02756 .48538 L
s
.02381 .04776 m
.02756 .04776 L
s
.02381 .01129 m
.02756 .01129 L
s
.02381 .55832 m
.02756 .55832 L
s
.02381 .59479 m
.02756 .59479 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.02 w
.02381 .01472 Mdot
.07101 .08459 Mdot
.14765 .18254 Mdot
.33089 .35548 Mdot
.51749 .46882 Mdot
.73881 .55175 Mdot
.97619 .60332 Mdot
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup]
Graphics["<<>>"]
;[o]
-Graphics-
:[font = input; preserveAspect; startGroup]
Show[pOil,p1new]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.02381 0.000238 0.014715 0.059595 [
[.2619 .00222 -12 -9 ]
[.2619 .00222 12 0 ]
[.5 .00222 -12 -9 ]
[.5 .00222 12 0 ]
[.7381 .00222 -12 -9 ]
[.7381 .00222 12 0 ]
[.97619 .00222 -12 -9 ]
[.97619 .00222 12 0 ]
[.01131 .1339 -6 -4.5 ]
[.01131 .1339 0 4.5 ]
[.01131 .25309 -6 -4.5 ]
[.01131 .25309 0 4.5 ]
[.01131 .37228 -6 -4.5 ]
[.01131 .37228 0 4.5 ]
[.01131 .49147 -6 -4.5 ]
[.01131 .49147 0 4.5 ]
[.01131 .61066 -12 -4.5 ]
[.01131 .61066 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.2619 .01472 m
.2619 .02097 L
s
[(1000)] .2619 .00222 0 1 Mshowa
.5 .01472 m
.5 .02097 L
s
[(2000)] .5 .00222 0 1 Mshowa
.7381 .01472 m
.7381 .02097 L
s
[(3000)] .7381 .00222 0 1 Mshowa
.97619 .01472 m
.97619 .02097 L
s
[(4000)] .97619 .00222 0 1 Mshowa
.125 Mabswid
.07143 .01472 m
.07143 .01847 L
s
.11905 .01472 m
.11905 .01847 L
s
.16667 .01472 m
.16667 .01847 L
s
.21429 .01472 m
.21429 .01847 L
s
.30952 .01472 m
.30952 .01847 L
s
.35714 .01472 m
.35714 .01847 L
s
.40476 .01472 m
.40476 .01847 L
s
.45238 .01472 m
.45238 .01847 L
s
.54762 .01472 m
.54762 .01847 L
s
.59524 .01472 m
.59524 .01847 L
s
.64286 .01472 m
.64286 .01847 L
s
.69048 .01472 m
.69048 .01847 L
s
.78571 .01472 m
.78571 .01847 L
s
.83333 .01472 m
.83333 .01847 L
s
.88095 .01472 m
.88095 .01847 L
s
.92857 .01472 m
.92857 .01847 L
s
.25 Mabswid
0 .01472 m
1 .01472 L
s
.02381 .1339 m
.03006 .1339 L
s
[(2)] .01131 .1339 1 0 Mshowa
.02381 .25309 m
.03006 .25309 L
s
[(4)] .01131 .25309 1 0 Mshowa
.02381 .37228 m
.03006 .37228 L
s
[(6)] .01131 .37228 1 0 Mshowa
.02381 .49147 m
.03006 .49147 L
s
[(8)] .01131 .49147 1 0 Mshowa
.02381 .61066 m
.03006 .61066 L
s
[(10)] .01131 .61066 1 0 Mshowa
.125 Mabswid
.02381 .04451 m
.02756 .04451 L
s
.02381 .07431 m
.02756 .07431 L
s
.02381 .10411 m
.02756 .10411 L
s
.02381 .1637 m
.02756 .1637 L
s
.02381 .1935 m
.02756 .1935 L
s
.02381 .2233 m
.02756 .2233 L
s
.02381 .28289 m
.02756 .28289 L
s
.02381 .31269 m
.02756 .31269 L
s
.02381 .34249 m
.02756 .34249 L
s
.02381 .40208 m
.02756 .40208 L
s
.02381 .43188 m
.02756 .43188 L
s
.02381 .46167 m
.02756 .46167 L
s
.02381 .52127 m
.02756 .52127 L
s
.02381 .55107 m
.02756 .55107 L
s
.02381 .58086 m
.02756 .58086 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.5 Mabswid
.02381 .07711 m
.06244 .15699 L
.10458 .23061 L
.14415 .2889 L
.18221 .33656 L
.22272 .37963 L
.26171 .41475 L
.30316 .44631 L
.34309 .47196 L
.3815 .49292 L
.42237 .51181 L
.46172 .5272 L
.49955 .53979 L
.53984 .55118 L
.57861 .56048 L
.61984 .56884 L
.65954 .57565 L
.69774 .58121 L
.73838 .58623 L
.77751 .59032 L
.81909 .59399 L
.85916 .59698 L
.89771 .59942 L
.93871 .60162 L
.97619 .60332 L
s
.02 w
.02381 .07711 Mdot
.05077 .1342 Mdot
.09455 .21424 Mdot
.19922 .35554 Mdot
.30581 .44815 Mdot
.43222 .51591 Mdot
.56782 .55804 Mdot
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
Graphics["<<>>"]
;[o]
-Graphics-
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
Now we determine the times at which our OilSlick[t] function is at the IO(t + 10)  times.
:[font = input; preserveAspect; startGroup]
timeL = Table[t/.FindRoot[IL[[i]]== OilSlick[t],
		{t,300}][[1]], {i,1, Length[IO]}];
:[font = message; inactive; preserveAspect; endGroup; endGroup]
General::spell1: 
   Possible spelling error: new symbol name "timeL"
     is similar to existing symbol "time".
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
And we list these times, t, and the corresponding OilSlick[t] values.
:[font = input; preserveAspect; startGroup]
newLdata = Table[{timeL[[i]],IL[[i]]},{i,1, Length[IO]}]
:[font = message; inactive; preserveAspect]
General::spell1: 
   Possible spelling error: new symbol name 
    "newLdata" is similar to existing symbol 
    "newdata".
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{{9.999622926568157, 1.135999999999999}, 
 
  {123.3039707961113, 2.084999999999999}, 
 
  {307.2157259492715, 3.415}, 
 
  {746.7520263255938, 5.761999999999999}, 
 
  {1194.593588511074, 7.301}, 
 
  {1725.227321627416, 8.426}, 
 
  {2294.627743032604, 9.125999999999999}}
;[o]
{{9.99962, 1.136}, {123.304, 2.085}, 
 
  {307.216, 3.415}, {746.752, 5.762}, 
 
  {1194.59, 7.301}, {1725.23, 8.426}, 
 
  {2294.63, 9.126}}
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
with a plot to follow of the IO(t+10) data over the function OilSLick[t].
:[font = input; preserveAspect; startGroup]
p1Lnew = ListPlot[newLdata,PlotStyle->{PointSize[.02]}]
:[font = message; inactive; preserveAspect]
General::spell1: 
   Possible spelling error: new symbol name "p1Lnew"
     is similar to existing symbol "p1new".
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.019641 0.000417 -0.068971 0.073668 [
[.22807 .06586 -9 -9 ]
[.22807 .06586 9 0 ]
[.43651 .06586 -12 -9 ]
[.43651 .06586 12 0 ]
[.64494 .06586 -12 -9 ]
[.64494 .06586 12 0 ]
[.85337 .06586 -12 -9 ]
[.85337 .06586 12 0 ]
[.00714 .2257 -6 -4.5 ]
[.00714 .2257 0 4.5 ]
[.00714 .37303 -6 -4.5 ]
[.00714 .37303 0 4.5 ]
[.00714 .52037 -6 -4.5 ]
[.00714 .52037 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.22807 .07836 m
.22807 .08461 L
s
[(500)] .22807 .06586 0 1 Mshowa
.43651 .07836 m
.43651 .08461 L
s
[(1000)] .43651 .06586 0 1 Mshowa
.64494 .07836 m
.64494 .08461 L
s
[(1500)] .64494 .06586 0 1 Mshowa
.85337 .07836 m
.85337 .08461 L
s
[(2000)] .85337 .06586 0 1 Mshowa
.125 Mabswid
.06133 .07836 m
.06133 .08211 L
s
.10301 .07836 m
.10301 .08211 L
s
.1447 .07836 m
.1447 .08211 L
s
.18639 .07836 m
.18639 .08211 L
s
.26976 .07836 m
.26976 .08211 L
s
.31145 .07836 m
.31145 .08211 L
s
.35313 .07836 m
.35313 .08211 L
s
.39482 .07836 m
.39482 .08211 L
s
.47819 .07836 m
.47819 .08211 L
s
.51988 .07836 m
.51988 .08211 L
s
.56157 .07836 m
.56157 .08211 L
s
.60325 .07836 m
.60325 .08211 L
s
.68662 .07836 m
.68662 .08211 L
s
.72831 .07836 m
.72831 .08211 L
s
.77 .07836 m
.77 .08211 L
s
.81168 .07836 m
.81168 .08211 L
s
.89506 .07836 m
.89506 .08211 L
s
.93674 .07836 m
.93674 .08211 L
s
.97843 .07836 m
.97843 .08211 L
s
.25 Mabswid
0 .07836 m
1 .07836 L
s
.01964 .2257 m
.02589 .2257 L
s
[(4)] .00714 .2257 1 0 Mshowa
.01964 .37303 m
.02589 .37303 L
s
[(6)] .00714 .37303 1 0 Mshowa
.01964 .52037 m
.02589 .52037 L
s
[(8)] .00714 .52037 1 0 Mshowa
.125 Mabswid
.01964 .1152 m
.02339 .1152 L
s
.01964 .15203 m
.02339 .15203 L
s
.01964 .18887 m
.02339 .18887 L
s
.01964 .26253 m
.02339 .26253 L
s
.01964 .29937 m
.02339 .29937 L
s
.01964 .3362 m
.02339 .3362 L
s
.01964 .40987 m
.02339 .40987 L
s
.01964 .4467 m
.02339 .4467 L
s
.01964 .48354 m
.02339 .48354 L
s
.01964 .04153 m
.02339 .04153 L
s
.01964 .0047 m
.02339 .0047 L
s
.01964 .5572 m
.02339 .5572 L
s
.01964 .59404 m
.02339 .59404 L
s
.25 Mabswid
.01964 0 m
.01964 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.02 w
.02381 .01472 Mdot
.07104 .08463 Mdot
.14771 .1826 Mdot
.33094 .3555 Mdot
.51763 .46888 Mdot
.73883 .55175 Mdot
.97619 .60332 Mdot
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
Graphics["<<>>"]
;[o]
-Graphics-
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
The following plot shows that the initial and 10 minute later data fit the model very well.
:[font = input; preserveAspect; startGroup]
Show[pOil,p1new,p1Lnew]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.02381 0.000238 0.014715 0.059595 [
[.2619 .00222 -12 -9 ]
[.2619 .00222 12 0 ]
[.5 .00222 -12 -9 ]
[.5 .00222 12 0 ]
[.7381 .00222 -12 -9 ]
[.7381 .00222 12 0 ]
[.97619 .00222 -12 -9 ]
[.97619 .00222 12 0 ]
[.01131 .1339 -6 -4.5 ]
[.01131 .1339 0 4.5 ]
[.01131 .25309 -6 -4.5 ]
[.01131 .25309 0 4.5 ]
[.01131 .37228 -6 -4.5 ]
[.01131 .37228 0 4.5 ]
[.01131 .49147 -6 -4.5 ]
[.01131 .49147 0 4.5 ]
[.01131 .61066 -12 -4.5 ]
[.01131 .61066 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.2619 .01472 m
.2619 .02097 L
s
[(1000)] .2619 .00222 0 1 Mshowa
.5 .01472 m
.5 .02097 L
s
[(2000)] .5 .00222 0 1 Mshowa
.7381 .01472 m
.7381 .02097 L
s
[(3000)] .7381 .00222 0 1 Mshowa
.97619 .01472 m
.97619 .02097 L
s
[(4000)] .97619 .00222 0 1 Mshowa
.125 Mabswid
.07143 .01472 m
.07143 .01847 L
s
.11905 .01472 m
.11905 .01847 L
s
.16667 .01472 m
.16667 .01847 L
s
.21429 .01472 m
.21429 .01847 L
s
.30952 .01472 m
.30952 .01847 L
s
.35714 .01472 m
.35714 .01847 L
s
.40476 .01472 m
.40476 .01847 L
s
.45238 .01472 m
.45238 .01847 L
s
.54762 .01472 m
.54762 .01847 L
s
.59524 .01472 m
.59524 .01847 L
s
.64286 .01472 m
.64286 .01847 L
s
.69048 .01472 m
.69048 .01847 L
s
.78571 .01472 m
.78571 .01847 L
s
.83333 .01472 m
.83333 .01847 L
s
.88095 .01472 m
.88095 .01847 L
s
.92857 .01472 m
.92857 .01847 L
s
.25 Mabswid
0 .01472 m
1 .01472 L
s
.02381 .1339 m
.03006 .1339 L
s
[(2)] .01131 .1339 1 0 Mshowa
.02381 .25309 m
.03006 .25309 L
s
[(4)] .01131 .25309 1 0 Mshowa
.02381 .37228 m
.03006 .37228 L
s
[(6)] .01131 .37228 1 0 Mshowa
.02381 .49147 m
.03006 .49147 L
s
[(8)] .01131 .49147 1 0 Mshowa
.02381 .61066 m
.03006 .61066 L
s
[(10)] .01131 .61066 1 0 Mshowa
.125 Mabswid
.02381 .04451 m
.02756 .04451 L
s
.02381 .07431 m
.02756 .07431 L
s
.02381 .10411 m
.02756 .10411 L
s
.02381 .1637 m
.02756 .1637 L
s
.02381 .1935 m
.02756 .1935 L
s
.02381 .2233 m
.02756 .2233 L
s
.02381 .28289 m
.02756 .28289 L
s
.02381 .31269 m
.02756 .31269 L
s
.02381 .34249 m
.02756 .34249 L
s
.02381 .40208 m
.02756 .40208 L
s
.02381 .43188 m
.02756 .43188 L
s
.02381 .46167 m
.02756 .46167 L
s
.02381 .52127 m
.02756 .52127 L
s
.02381 .55107 m
.02756 .55107 L
s
.02381 .58086 m
.02756 .58086 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.5 Mabswid
.02381 .07711 m
.06244 .15699 L
.10458 .23061 L
.14415 .2889 L
.18221 .33656 L
.22272 .37963 L
.26171 .41475 L
.30316 .44631 L
.34309 .47196 L
.3815 .49292 L
.42237 .51181 L
.46172 .5272 L
.49955 .53979 L
.53984 .55118 L
.57861 .56048 L
.61984 .56884 L
.65954 .57565 L
.69774 .58121 L
.73838 .58623 L
.77751 .59032 L
.81909 .59399 L
.85916 .59698 L
.89771 .59942 L
.93871 .60162 L
.97619 .60332 L
s
.02 w
.02381 .07711 Mdot
.05077 .1342 Mdot
.09455 .21424 Mdot
.19922 .35554 Mdot
.30581 .44815 Mdot
.43222 .51591 Mdot
.56782 .55804 Mdot
.02619 .08241 Mdot
.05317 .13897 Mdot
.09696 .21823 Mdot
.20161 .3581 Mdot
.30824 .44981 Mdot
.43458 .51686 Mdot
.57015 .55857 Mdot
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
Graphics["<<>>"]
;[o]
-Graphics-
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
And this zoom plot confirms the fit even more.
:[font = input; preserveAspect; startGroup]
Show[pOil,p1new,p1Lnew,PlotRange->{{600,800},{0,10}}]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
-3 0.005 0 0.061803 [
[.125 -0.0125 -9 -9 ]
[.125 -0.0125 9 0 ]
[.25 -0.0125 -9 -9 ]
[.25 -0.0125 9 0 ]
[.375 -0.0125 -9 -9 ]
[.375 -0.0125 9 0 ]
[.5 -0.0125 -9 -9 ]
[.5 -0.0125 9 0 ]
[.625 -0.0125 -9 -9 ]
[.625 -0.0125 9 0 ]
[.75 -0.0125 -9 -9 ]
[.75 -0.0125 9 0 ]
[.875 -0.0125 -9 -9 ]
[.875 -0.0125 9 0 ]
[1 -0.0125 -9 -9 ]
[1 -0.0125 9 0 ]
[-0.0125 .12361 -6 -4.5 ]
[-0.0125 .12361 0 4.5 ]
[-0.0125 .24721 -6 -4.5 ]
[-0.0125 .24721 0 4.5 ]
[-0.0125 .37082 -6 -4.5 ]
[-0.0125 .37082 0 4.5 ]
[-0.0125 .49443 -6 -4.5 ]
[-0.0125 .49443 0 4.5 ]
[-0.0125 .61803 -12 -4.5 ]
[-0.0125 .61803 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.125 0 m
.125 .00625 L
s
[(625)] .125 -0.0125 0 1 Mshowa
.25 0 m
.25 .00625 L
s
[(650)] .25 -0.0125 0 1 Mshowa
.375 0 m
.375 .00625 L
s
[(675)] .375 -0.0125 0 1 Mshowa
.5 0 m
.5 .00625 L
s
[(700)] .5 -0.0125 0 1 Mshowa
.625 0 m
.625 .00625 L
s
[(725)] .625 -0.0125 0 1 Mshowa
.75 0 m
.75 .00625 L
s
[(750)] .75 -0.0125 0 1 Mshowa
.875 0 m
.875 .00625 L
s
[(775)] .875 -0.0125 0 1 Mshowa
1 0 m
1 .00625 L
s
[(800)] 1 -0.0125 0 1 Mshowa
.125 Mabswid
.025 0 m
.025 .00375 L
s
.05 0 m
.05 .00375 L
s
.075 0 m
.075 .00375 L
s
.1 0 m
.1 .00375 L
s
.15 0 m
.15 .00375 L
s
.175 0 m
.175 .00375 L
s
.2 0 m
.2 .00375 L
s
.225 0 m
.225 .00375 L
s
.275 0 m
.275 .00375 L
s
.3 0 m
.3 .00375 L
s
.325 0 m
.325 .00375 L
s
.35 0 m
.35 .00375 L
s
.4 0 m
.4 .00375 L
s
.425 0 m
.425 .00375 L
s
.45 0 m
.45 .00375 L
s
.475 0 m
.475 .00375 L
s
.525 0 m
.525 .00375 L
s
.55 0 m
.55 .00375 L
s
.575 0 m
.575 .00375 L
s
.6 0 m
.6 .00375 L
s
.65 0 m
.65 .00375 L
s
.675 0 m
.675 .00375 L
s
.7 0 m
.7 .00375 L
s
.725 0 m
.725 .00375 L
s
.775 0 m
.775 .00375 L
s
.8 0 m
.8 .00375 L
s
.825 0 m
.825 .00375 L
s
.85 0 m
.85 .00375 L
s
.9 0 m
.9 .00375 L
s
.925 0 m
.925 .00375 L
s
.95 0 m
.95 .00375 L
s
.975 0 m
.975 .00375 L
s
.25 Mabswid
0 0 m
1 0 L
s
0 .12361 m
.00625 .12361 L
s
[(2)] -0.0125 .12361 1 0 Mshowa
0 .24721 m
.00625 .24721 L
s
[(4)] -0.0125 .24721 1 0 Mshowa
0 .37082 m
.00625 .37082 L
s
[(6)] -0.0125 .37082 1 0 Mshowa
0 .49443 m
.00625 .49443 L
s
[(8)] -0.0125 .49443 1 0 Mshowa
0 .61803 m
.00625 .61803 L
s
[(10)] -0.0125 .61803 1 0 Mshowa
.125 Mabswid
0 .0309 m
.00375 .0309 L
s
0 .0618 m
.00375 .0618 L
s
0 .09271 m
.00375 .09271 L
s
0 .15451 m
.00375 .15451 L
s
0 .18541 m
.00375 .18541 L
s
0 .21631 m
.00375 .21631 L
s
0 .27812 m
.00375 .27812 L
s
0 .30902 m
.00375 .30902 L
s
0 .33992 m
.00375 .33992 L
s
0 .40172 m
.00375 .40172 L
s
0 .43262 m
.00375 .43262 L
s
0 .46353 m
.00375 .46353 L
s
0 .52533 m
.00375 .52533 L
s
0 .55623 m
.00375 .55623 L
s
0 .58713 m
.00375 .58713 L
s
.25 Mabswid
0 0 m
0 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.5 Mabswid
0 .31359 m
.32637 .33378 L
s
.32637 .33378 m
1 .36915 L
s
.02 w
.68352 .35345 Mdot
.73376 .35611 Mdot
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
Graphics["<<>>"]
;[o]
-Graphics-
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
Finally here is what the OilSlick model predicts the 10 minutes later size of the slick should be:
:[font = input; preserveAspect; startGroup]
M = OilSlick[time+10]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{1.136003338898604, 2.084527969101046, 
 
  3.414244669957872, 5.761793027022001, 
 
  7.300423388531857, 8.426176837348537, 
 
  9.126184603578783}
;[o]
{1.136, 2.08453, 3.41424, 5.76179, 7.30042, 8.42618, 
 
  9.12618}
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
We compare this with the 10 minutes later data,
:[font = input; preserveAspect; startGroup]
IL
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{1.135999999999999, 2.084999999999999, 3.415, 
 
  5.761999999999999, 7.301, 8.426, 9.125999999999999}
;[o]
{1.136, 2.085, 3.415, 5.762, 7.301, 8.426, 9.126}
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
even differencing them at each increment to see how close our model is.
:[font = input; preserveAspect; endGroup; endGroup]
Table[M[[i]] - IL[[i]],{i,1, Length[M]}];
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
We now offer an alternative solution brought about by class discussion of several groups of students in which average growth rate is used to estimate the time intervals between initial observations. 

In this approach we attempt to estimate the growth rate (average growth rate) over each 10 minute interval and use this growth rate to predict the amount of time between the end observation at one interval and the initial observation at the next interval by dividing this average growth rate into the difference in the growth from one observation to the next.
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
We enter the  Difference  between Initial Observation Data (IO) and the ten minute later Observation Data (IL) (divided by 10 minutes to get the average per minute change in the size of the oil slick over the ten minute interval.)  
:[font = input; preserveAspect; startGroup]
IDAve = Table[(IL[[i]] - IO[[i]])/10,{i,1,Length[IO]}]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{0.008899999999999996, 0.008000000000000007, 
 
  0.006700000000000017, 0.004299999999999926, 
 
  0.002800000000000046, 0.001600000000000001, 
 
  0.0008999999999998564}
;[o]
{0.0089, 0.008, 0.0067, 0.0043, 0.0028, 0.0016, 
 
  0.0009}
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
We divide the average growth rate into the actual growth  between the later observation and the next initial observation  to get an estimate of the time between  these two observations:
:[font = input; preserveAspect; endGroup]
dataTime = Table[(IO[[i+1]] - IL[[i]])/IDAve[[i]] ,
	{i,1, Length[IO]-1}];
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
We convert these times to actual times from t = 0 when the first initial observation is taken, by adding 10 minute intervals - to account for the time interval of the measurements.
:[font = input; preserveAspect]
NewTime = Table[
Sum[dataTime[[i]] + 10,{i,1,j}],
		{j,1, Length[dataTime]}];
:[font = input; preserveAspect; endGroup]
NewTime = PrependTo[NewTime,0];
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
Here we determine the (time, size) data
:[font = input; preserveAspect; startGroup]
goodData = Table[{NewTime[[i]],IO[[i]]},{i,1, Length[IO]}]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{{0, 1.046999999999999}, 
 
  {107.6404494382023, 2.004999999999999}, 
 
  {275.5154494382021, 3.347999999999999}, 
 
  {629.3960464531267, 5.719}, 
 
  {990.791395290342, 7.272999999999999}, 
 
  {1396.862823861763, 8.41}, 
 
  {1838.737823861763, 9.117}}
;[o]
{{0, 1.047}, {107.64, 2.005}, {275.515, 3.348}, 
 
  {629.396, 5.719}, {990.791, 7.273}, 
 
  {1396.86, 8.41}, {1838.74, 9.117}}
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
and plot the revised data to compare with the function we obtained from our first approach to the problem.
:[font = input; preserveAspect; startGroup]
p1Good = ListPlot[goodData,PlotStyle->{PointSize[.02]}]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.02381 0.000518 -0.06165 0.072937 [
[.1533 .07172 -9 -9 ]
[.1533 .07172 9 0 ]
[.28279 .07172 -9 -9 ]
[.28279 .07172 9 0 ]
[.41227 .07172 -9 -9 ]
[.41227 .07172 9 0 ]
[.54176 .07172 -12 -9 ]
[.54176 .07172 12 0 ]
[.67125 .07172 -12 -9 ]
[.67125 .07172 12 0 ]
[.80074 .07172 -12 -9 ]
[.80074 .07172 12 0 ]
[.93023 .07172 -12 -9 ]
[.93023 .07172 12 0 ]
[.01131 .2301 -6 -4.5 ]
[.01131 .2301 0 4.5 ]
[.01131 .37597 -6 -4.5 ]
[.01131 .37597 0 4.5 ]
[.01131 .52185 -6 -4.5 ]
[.01131 .52185 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.1533 .08422 m
.1533 .09047 L
s
[(250)] .1533 .07172 0 1 Mshowa
.28279 .08422 m
.28279 .09047 L
s
[(500)] .28279 .07172 0 1 Mshowa
.41227 .08422 m
.41227 .09047 L
s
[(750)] .41227 .07172 0 1 Mshowa
.54176 .08422 m
.54176 .09047 L
s
[(1000)] .54176 .07172 0 1 Mshowa
.67125 .08422 m
.67125 .09047 L
s
[(1250)] .67125 .07172 0 1 Mshowa
.80074 .08422 m
.80074 .09047 L
s
[(1500)] .80074 .07172 0 1 Mshowa
.93023 .08422 m
.93023 .09047 L
s
[(1750)] .93023 .07172 0 1 Mshowa
.125 Mabswid
.04971 .08422 m
.04971 .08797 L
s
.0756 .08422 m
.0756 .08797 L
s
.1015 .08422 m
.1015 .08797 L
s
.1274 .08422 m
.1274 .08797 L
s
.1792 .08422 m
.1792 .08797 L
s
.20509 .08422 m
.20509 .08797 L
s
.23099 .08422 m
.23099 .08797 L
s
.25689 .08422 m
.25689 .08797 L
s
.30868 .08422 m
.30868 .08797 L
s
.33458 .08422 m
.33458 .08797 L
s
.36048 .08422 m
.36048 .08797 L
s
.38638 .08422 m
.38638 .08797 L
s
.43817 .08422 m
.43817 .08797 L
s
.46407 .08422 m
.46407 .08797 L
s
.48997 .08422 m
.48997 .08797 L
s
.51587 .08422 m
.51587 .08797 L
s
.56766 .08422 m
.56766 .08797 L
s
.59356 .08422 m
.59356 .08797 L
s
.61946 .08422 m
.61946 .08797 L
s
.64535 .08422 m
.64535 .08797 L
s
.69715 .08422 m
.69715 .08797 L
s
.72305 .08422 m
.72305 .08797 L
s
.74894 .08422 m
.74894 .08797 L
s
.77484 .08422 m
.77484 .08797 L
s
.82664 .08422 m
.82664 .08797 L
s
.85254 .08422 m
.85254 .08797 L
s
.87843 .08422 m
.87843 .08797 L
s
.90433 .08422 m
.90433 .08797 L
s
.95613 .08422 m
.95613 .08797 L
s
.98202 .08422 m
.98202 .08797 L
s
.25 Mabswid
0 .08422 m
1 .08422 L
s
.02381 .2301 m
.03006 .2301 L
s
[(4)] .01131 .2301 1 0 Mshowa
.02381 .37597 m
.03006 .37597 L
s
[(6)] .01131 .37597 1 0 Mshowa
.02381 .52185 m
.03006 .52185 L
s
[(8)] .01131 .52185 1 0 Mshowa
.125 Mabswid
.02381 .12069 m
.02756 .12069 L
s
.02381 .15716 m
.02756 .15716 L
s
.02381 .19363 m
.02756 .19363 L
s
.02381 .26657 m
.02756 .26657 L
s
.02381 .30304 m
.02756 .30304 L
s
.02381 .3395 m
.02756 .3395 L
s
.02381 .41244 m
.02756 .41244 L
s
.02381 .44891 m
.02756 .44891 L
s
.02381 .48538 m
.02756 .48538 L
s
.02381 .04776 m
.02756 .04776 L
s
.02381 .01129 m
.02756 .01129 L
s
.02381 .55832 m
.02756 .55832 L
s
.02381 .59479 m
.02756 .59479 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.02 w
.02381 .01472 Mdot
.07956 .08459 Mdot
.16651 .18254 Mdot
.34981 .35548 Mdot
.53699 .46882 Mdot
.74732 .55175 Mdot
.97619 .60332 Mdot
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup]
Graphics["<<>>"]
;[o]
-Graphics-
:[font = input; preserveAspect; startGroup]
Show[pOil,p1Good]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.02381 0.000238 0.014715 0.059595 [
[.2619 .00222 -12 -9 ]
[.2619 .00222 12 0 ]
[.5 .00222 -12 -9 ]
[.5 .00222 12 0 ]
[.7381 .00222 -12 -9 ]
[.7381 .00222 12 0 ]
[.97619 .00222 -12 -9 ]
[.97619 .00222 12 0 ]
[.01131 .1339 -6 -4.5 ]
[.01131 .1339 0 4.5 ]
[.01131 .25309 -6 -4.5 ]
[.01131 .25309 0 4.5 ]
[.01131 .37228 -6 -4.5 ]
[.01131 .37228 0 4.5 ]
[.01131 .49147 -6 -4.5 ]
[.01131 .49147 0 4.5 ]
[.01131 .61066 -12 -4.5 ]
[.01131 .61066 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.2619 .01472 m
.2619 .02097 L
s
[(1000)] .2619 .00222 0 1 Mshowa
.5 .01472 m
.5 .02097 L
s
[(2000)] .5 .00222 0 1 Mshowa
.7381 .01472 m
.7381 .02097 L
s
[(3000)] .7381 .00222 0 1 Mshowa
.97619 .01472 m
.97619 .02097 L
s
[(4000)] .97619 .00222 0 1 Mshowa
.125 Mabswid
.07143 .01472 m
.07143 .01847 L
s
.11905 .01472 m
.11905 .01847 L
s
.16667 .01472 m
.16667 .01847 L
s
.21429 .01472 m
.21429 .01847 L
s
.30952 .01472 m
.30952 .01847 L
s
.35714 .01472 m
.35714 .01847 L
s
.40476 .01472 m
.40476 .01847 L
s
.45238 .01472 m
.45238 .01847 L
s
.54762 .01472 m
.54762 .01847 L
s
.59524 .01472 m
.59524 .01847 L
s
.64286 .01472 m
.64286 .01847 L
s
.69048 .01472 m
.69048 .01847 L
s
.78571 .01472 m
.78571 .01847 L
s
.83333 .01472 m
.83333 .01847 L
s
.88095 .01472 m
.88095 .01847 L
s
.92857 .01472 m
.92857 .01847 L
s
.25 Mabswid
0 .01472 m
1 .01472 L
s
.02381 .1339 m
.03006 .1339 L
s
[(2)] .01131 .1339 1 0 Mshowa
.02381 .25309 m
.03006 .25309 L
s
[(4)] .01131 .25309 1 0 Mshowa
.02381 .37228 m
.03006 .37228 L
s
[(6)] .01131 .37228 1 0 Mshowa
.02381 .49147 m
.03006 .49147 L
s
[(8)] .01131 .49147 1 0 Mshowa
.02381 .61066 m
.03006 .61066 L
s
[(10)] .01131 .61066 1 0 Mshowa
.125 Mabswid
.02381 .04451 m
.02756 .04451 L
s
.02381 .07431 m
.02756 .07431 L
s
.02381 .10411 m
.02756 .10411 L
s
.02381 .1637 m
.02756 .1637 L
s
.02381 .1935 m
.02756 .1935 L
s
.02381 .2233 m
.02756 .2233 L
s
.02381 .28289 m
.02756 .28289 L
s
.02381 .31269 m
.02756 .31269 L
s
.02381 .34249 m
.02756 .34249 L
s
.02381 .40208 m
.02756 .40208 L
s
.02381 .43188 m
.02756 .43188 L
s
.02381 .46167 m
.02756 .46167 L
s
.02381 .52127 m
.02756 .52127 L
s
.02381 .55107 m
.02756 .55107 L
s
.02381 .58086 m
.02756 .58086 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.5 Mabswid
.02381 .07711 m
.06244 .15699 L
.10458 .23061 L
.14415 .2889 L
.18221 .33656 L
.22272 .37963 L
.26171 .41475 L
.30316 .44631 L
.34309 .47196 L
.3815 .49292 L
.42237 .51181 L
.46172 .5272 L
.49955 .53979 L
.53984 .55118 L
.57861 .56048 L
.61984 .56884 L
.65954 .57565 L
.69774 .58121 L
.73838 .58623 L
.77751 .59032 L
.81909 .59399 L
.85916 .59698 L
.89771 .59942 L
.93871 .60162 L
.97619 .60332 L
s
.02 w
.02381 .07711 Mdot
.04944 .1342 Mdot
.08941 .21424 Mdot
.17367 .35554 Mdot
.25971 .44815 Mdot
.3564 .51591 Mdot
.4616 .55804 Mdot
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
Graphics["<<>>"]
;[o]
-Graphics-
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
Oops this would appear to show that we have calculated our between observation time intervals incorrectly, in particular we have used a bigger growth rate over the entire interval between  observations.  
:[font = input; preserveAspect; startGroup]
IDAve = Table[(IL[[i]] - IO[[i]])/10,{i,1,Length[IO]}]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{0.008899999999999996, 0.008000000000000007, 
 
  0.006700000000000017, 0.004299999999999926, 
 
  0.002800000000000046, 0.001600000000000001, 
 
  0.0008999999999998564}
;[o]
{0.0089, 0.008, 0.0067, 0.0043, 0.0028, 0.0016, 
 
  0.0009}
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
We need to estimate the growth rate in the interval between observation intervals and so we take the average of the growth rates from consecutive intervals to determine the time it takes between observation intervals.
:[font = input; preserveAspect; startGroup]
IDNewAve = Table[((IL[[i]] - IO[[i]])/10 + 
	(IL[[i+1]] - IO[[i+1]])/10)/2,{i,1,Length[IO]-1}]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{0.008450000000000002, 0.007350000000000012, 
 
  0.005499999999999971, 0.003549999999999986, 
 
  0.002200000000000024, 0.001249999999999928}
;[o]
{0.00845, 0.00735, 0.0055, 0.00355, 0.0022, 0.00125}
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
We divide the average growth rate into the actual growth  between the later observation and the next initial observation  to get an estimate of the time between  these two observations:
:[font = input; preserveAspect; startGroup]
dataNewTime = Table[(IO[[i+1]] - IL[[i]])/IDNewAve[[i]] ,
	{i,1, Length[IO]-1}]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{102.8402366863905, 171.8367346938772, 
 
  418.909090909093, 425.633802816903, 
 
  504.0909090909035, 552.800000000032}
;[o]
{102.84, 171.837, 418.909, 425.634, 504.091, 552.8}
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
We convert these times to actual times from t = 0 when the first initial observation is taken, by adding 10 minute intervals - to account for the time interval of the measurements.
:[font = input; preserveAspect]
NewNewTime = Table[
Sum[dataNewTime[[i]] + 10,{i,1,j}],
		{j,1, Length[dataNewTime]}];
:[font = input; preserveAspect; endGroup]
NewNewTime = PrependTo[NewNewTime,0];
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
Here we determine the (time, size) data
:[font = input; preserveAspect; startGroup]
goodNewData = Table[{NewNewTime[[i]],IO[[i]]},
		{i,1, Length[IO]}]
:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
{{0, 1.046999999999999}, 
 
  {112.8402366863904, 2.004999999999999}, 
 
  {294.6769713802678, 3.347999999999999}, 
 
  {723.5860622893608, 5.719}, 
 
  {1159.219865106263, 7.272999999999999}, 
 
  {1673.310774197167, 8.41}, 
 
  {2236.110774197199, 9.117}}
;[o]
{{0, 1.047}, {112.84, 2.005}, {294.677, 3.348}, 
 
  {723.586, 5.719}, {1159.22, 7.273}, 
 
  {1673.31, 8.41}, {2236.11, 9.117}}
:[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup]
and plot the revised data to compare with the function we obtained from our first approach to the problem.
:[font = input; preserveAspect; startGroup]
p1NewGood = ListPlot[goodNewData,PlotStyle->
			{PointSize[.02]}]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.02381 0.000426 -0.06165 0.072937 [
[.23676 .07172 -9 -9 ]
[.23676 .07172 9 0 ]
[.44972 .07172 -12 -9 ]
[.44972 .07172 12 0 ]
[.66267 .07172 -12 -9 ]
[.66267 .07172 12 0 ]
[.87563 .07172 -12 -9 ]
[.87563 .07172 12 0 ]
[.01131 .2301 -6 -4.5 ]
[.01131 .2301 0 4.5 ]
[.01131 .37597 -6 -4.5 ]
[.01131 .37597 0 4.5 ]
[.01131 .52185 -6 -4.5 ]
[.01131 .52185 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.23676 .08422 m
.23676 .09047 L
s
[(500)] .23676 .07172 0 1 Mshowa
.44972 .08422 m
.44972 .09047 L
s
[(1000)] .44972 .07172 0 1 Mshowa
.66267 .08422 m
.66267 .09047 L
s
[(1500)] .66267 .07172 0 1 Mshowa
.87563 .08422 m
.87563 .09047 L
s
[(2000)] .87563 .07172 0 1 Mshowa
.125 Mabswid
.0664 .08422 m
.0664 .08797 L
s
.10899 .08422 m
.10899 .08797 L
s
.15158 .08422 m
.15158 .08797 L
s
.19417 .08422 m
.19417 .08797 L
s
.27936 .08422 m
.27936 .08797 L
s
.32195 .08422 m
.32195 .08797 L
s
.36454 .08422 m
.36454 .08797 L
s
.40713 .08422 m
.40713 .08797 L
s
.49231 .08422 m
.49231 .08797 L
s
.5349 .08422 m
.5349 .08797 L
s
.57749 .08422 m
.57749 .08797 L
s
.62008 .08422 m
.62008 .08797 L
s
.70526 .08422 m
.70526 .08797 L
s
.74786 .08422 m
.74786 .08797 L
s
.79045 .08422 m
.79045 .08797 L
s
.83304 .08422 m
.83304 .08797 L
s
.91822 .08422 m
.91822 .08797 L
s
.96081 .08422 m
.96081 .08797 L
s
.25 Mabswid
0 .08422 m
1 .08422 L
s
.02381 .2301 m
.03006 .2301 L
s
[(4)] .01131 .2301 1 0 Mshowa
.02381 .37597 m
.03006 .37597 L
s
[(6)] .01131 .37597 1 0 Mshowa
.02381 .52185 m
.03006 .52185 L
s
[(8)] .01131 .52185 1 0 Mshowa
.125 Mabswid
.02381 .12069 m
.02756 .12069 L
s
.02381 .15716 m
.02756 .15716 L
s
.02381 .19363 m
.02756 .19363 L
s
.02381 .26657 m
.02756 .26657 L
s
.02381 .30304 m
.02756 .30304 L
s
.02381 .3395 m
.02756 .3395 L
s
.02381 .41244 m
.02756 .41244 L
s
.02381 .44891 m
.02756 .44891 L
s
.02381 .48538 m
.02756 .48538 L
s
.02381 .04776 m
.02756 .04776 L
s
.02381 .01129 m
.02756 .01129 L
s
.02381 .55832 m
.02756 .55832 L
s
.02381 .59479 m
.02756 .59479 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.02 w
.02381 .01472 Mdot
.07187 .08459 Mdot
.14932 .18254 Mdot
.33199 .35548 Mdot
.51753 .46882 Mdot
.73649 .55175 Mdot
.97619 .60332 Mdot
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup]
Graphics["<<>>"]
;[o]
-Graphics-
:[font = input; preserveAspect; startGroup]
Show[pOil,p1NewGood]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.02381 0.000238 0.014715 0.059595 [
[.2619 .00222 -12 -9 ]
[.2619 .00222 12 0 ]
[.5 .00222 -12 -9 ]
[.5 .00222 12 0 ]
[.7381 .00222 -12 -9 ]
[.7381 .00222 12 0 ]
[.97619 .00222 -12 -9 ]
[.97619 .00222 12 0 ]
[.01131 .1339 -6 -4.5 ]
[.01131 .1339 0 4.5 ]
[.01131 .25309 -6 -4.5 ]
[.01131 .25309 0 4.5 ]
[.01131 .37228 -6 -4.5 ]
[.01131 .37228 0 4.5 ]
[.01131 .49147 -6 -4.5 ]
[.01131 .49147 0 4.5 ]
[.01131 .61066 -12 -4.5 ]
[.01131 .61066 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.2619 .01472 m
.2619 .02097 L
s
[(1000)] .2619 .00222 0 1 Mshowa
.5 .01472 m
.5 .02097 L
s
[(2000)] .5 .00222 0 1 Mshowa
.7381 .01472 m
.7381 .02097 L
s
[(3000)] .7381 .00222 0 1 Mshowa
.97619 .01472 m
.97619 .02097 L
s
[(4000)] .97619 .00222 0 1 Mshowa
.125 Mabswid
.07143 .01472 m
.07143 .01847 L
s
.11905 .01472 m
.11905 .01847 L
s
.16667 .01472 m
.16667 .01847 L
s
.21429 .01472 m
.21429 .01847 L
s
.30952 .01472 m
.30952 .01847 L
s
.35714 .01472 m
.35714 .01847 L
s
.40476 .01472 m
.40476 .01847 L
s
.45238 .01472 m
.45238 .01847 L
s
.54762 .01472 m
.54762 .01847 L
s
.59524 .01472 m
.59524 .01847 L
s
.64286 .01472 m
.64286 .01847 L
s
.69048 .01472 m
.69048 .01847 L
s
.78571 .01472 m
.78571 .01847 L
s
.83333 .01472 m
.83333 .01847 L
s
.88095 .01472 m
.88095 .01847 L
s
.92857 .01472 m
.92857 .01847 L
s
.25 Mabswid
0 .01472 m
1 .01472 L
s
.02381 .1339 m
.03006 .1339 L
s
[(2)] .01131 .1339 1 0 Mshowa
.02381 .25309 m
.03006 .25309 L
s
[(4)] .01131 .25309 1 0 Mshowa
.02381 .37228 m
.03006 .37228 L
s
[(6)] .01131 .37228 1 0 Mshowa
.02381 .49147 m
.03006 .49147 L
s
[(8)] .01131 .49147 1 0 Mshowa
.02381 .61066 m
.03006 .61066 L
s
[(10)] .01131 .61066 1 0 Mshowa
.125 Mabswid
.02381 .04451 m
.02756 .04451 L
s
.02381 .07431 m
.02756 .07431 L
s
.02381 .10411 m
.02756 .10411 L
s
.02381 .1637 m
.02756 .1637 L
s
.02381 .1935 m
.02756 .1935 L
s
.02381 .2233 m
.02756 .2233 L
s
.02381 .28289 m
.02756 .28289 L
s
.02381 .31269 m
.02756 .31269 L
s
.02381 .34249 m
.02756 .34249 L
s
.02381 .40208 m
.02756 .40208 L
s
.02381 .43188 m
.02756 .43188 L
s
.02381 .46167 m
.02756 .46167 L
s
.02381 .52127 m
.02756 .52127 L
s
.02381 .55107 m
.02756 .55107 L
s
.02381 .58086 m
.02756 .58086 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.5 Mabswid
.02381 .07711 m
.06244 .15699 L
.10458 .23061 L
.14415 .2889 L
.18221 .33656 L
.22272 .37963 L
.26171 .41475 L
.30316 .44631 L
.34309 .47196 L
.3815 .49292 L
.42237 .51181 L
.46172 .5272 L
.49955 .53979 L
.53984 .55118 L
.57861 .56048 L
.61984 .56884 L
.65954 .57565 L
.69774 .58121 L
.73838 .58623 L
.77751 .59032 L
.81909 .59399 L
.85916 .59698 L
.89771 .59942 L
.93871 .60162 L
.97619 .60332 L
s
.02 w
.02381 .07711 Mdot
.05068 .1342 Mdot
.09397 .21424 Mdot
.19609 .35554 Mdot
.29981 .44815 Mdot
.42222 .51591 Mdot
.55622 .55804 Mdot
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup; endGroup]
Graphics["<<>>"]
;[o]
-Graphics-
:[font = subsubsection; inactive; preserveAspect; endGroup]
That looks better and we have a reasonable model here.
:[font = subsection; inactive; Cclosed; preserveAspect; startGroup]
And now let us plot both models over the data:
:[font = input; preserveAspect; startGroup]
Show[pOil,p1new,p1Lnew,p1NewGood]
:[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174]
%!
%%Creator: Mathematica
%%AspectRatio: .61803 
MathPictureStart
/Mabs {
Mgmatrix idtransform
Mtmatrix dtransform
} bind def
/Mabsadd { Mabs
3 -1 roll add
3 1 roll add
exch } bind def
%% Graphics
/Courier findfont 10  scalefont  setfont
% Scaling calculations
0.02381 0.000238 0.014715 0.059595 [
[.2619 .00222 -12 -9 ]
[.2619 .00222 12 0 ]
[.5 .00222 -12 -9 ]
[.5 .00222 12 0 ]
[.7381 .00222 -12 -9 ]
[.7381 .00222 12 0 ]
[.97619 .00222 -12 -9 ]
[.97619 .00222 12 0 ]
[.01131 .1339 -6 -4.5 ]
[.01131 .1339 0 4.5 ]
[.01131 .25309 -6 -4.5 ]
[.01131 .25309 0 4.5 ]
[.01131 .37228 -6 -4.5 ]
[.01131 .37228 0 4.5 ]
[.01131 .49147 -6 -4.5 ]
[.01131 .49147 0 4.5 ]
[.01131 .61066 -12 -4.5 ]
[.01131 .61066 0 4.5 ]
[ 0 0 0 0 ]
[ 1 .61803 0 0 ]
] MathScale
% Start of Graphics
1 setlinecap
1 setlinejoin
newpath
0 g
.25 Mabswid
[ ] 0 setdash
.2619 .01472 m
.2619 .02097 L
s
[(1000)] .2619 .00222 0 1 Mshowa
.5 .01472 m
.5 .02097 L
s
[(2000)] .5 .00222 0 1 Mshowa
.7381 .01472 m
.7381 .02097 L
s
[(3000)] .7381 .00222 0 1 Mshowa
.97619 .01472 m
.97619 .02097 L
s
[(4000)] .97619 .00222 0 1 Mshowa
.125 Mabswid
.07143 .01472 m
.07143 .01847 L
s
.11905 .01472 m
.11905 .01847 L
s
.16667 .01472 m
.16667 .01847 L
s
.21429 .01472 m
.21429 .01847 L
s
.30952 .01472 m
.30952 .01847 L
s
.35714 .01472 m
.35714 .01847 L
s
.40476 .01472 m
.40476 .01847 L
s
.45238 .01472 m
.45238 .01847 L
s
.54762 .01472 m
.54762 .01847 L
s
.59524 .01472 m
.59524 .01847 L
s
.64286 .01472 m
.64286 .01847 L
s
.69048 .01472 m
.69048 .01847 L
s
.78571 .01472 m
.78571 .01847 L
s
.83333 .01472 m
.83333 .01847 L
s
.88095 .01472 m
.88095 .01847 L
s
.92857 .01472 m
.92857 .01847 L
s
.25 Mabswid
0 .01472 m
1 .01472 L
s
.02381 .1339 m
.03006 .1339 L
s
[(2)] .01131 .1339 1 0 Mshowa
.02381 .25309 m
.03006 .25309 L
s
[(4)] .01131 .25309 1 0 Mshowa
.02381 .37228 m
.03006 .37228 L
s
[(6)] .01131 .37228 1 0 Mshowa
.02381 .49147 m
.03006 .49147 L
s
[(8)] .01131 .49147 1 0 Mshowa
.02381 .61066 m
.03006 .61066 L
s
[(10)] .01131 .61066 1 0 Mshowa
.125 Mabswid
.02381 .04451 m
.02756 .04451 L
s
.02381 .07431 m
.02756 .07431 L
s
.02381 .10411 m
.02756 .10411 L
s
.02381 .1637 m
.02756 .1637 L
s
.02381 .1935 m
.02756 .1935 L
s
.02381 .2233 m
.02756 .2233 L
s
.02381 .28289 m
.02756 .28289 L
s
.02381 .31269 m
.02756 .31269 L
s
.02381 .34249 m
.02756 .34249 L
s
.02381 .40208 m
.02756 .40208 L
s
.02381 .43188 m
.02756 .43188 L
s
.02381 .46167 m
.02756 .46167 L
s
.02381 .52127 m
.02756 .52127 L
s
.02381 .55107 m
.02756 .55107 L
s
.02381 .58086 m
.02756 .58086 L
s
.25 Mabswid
.02381 0 m
.02381 .61803 L
s
0 0 m
1 0 L
1 .61803 L
0 .61803 L
closepath
clip
newpath
.5 Mabswid
.02381 .07711 m
.06244 .15699 L
.10458 .23061 L
.14415 .2889 L
.18221 .33656 L
.22272 .37963 L
.26171 .41475 L
.30316 .44631 L
.34309 .47196 L
.3815 .49292 L
.42237 .51181 L
.46172 .5272 L
.49955 .53979 L
.53984 .55118 L
.57861 .56048 L
.61984 .56884 L
.65954 .57565 L
.69774 .58121 L
.73838 .58623 L
.77751 .59032 L
.81909 .59399 L
.85916 .59698 L
.89771 .59942 L
.93871 .60162 L
.97619 .60332 L
s
.02 w
.02381 .07711 Mdot
.05077 .1342 Mdot
.09455 .21424 Mdot
.19922 .35554 Mdot
.30581 .44815 Mdot
.43222 .51591 Mdot
.56782 .55804 Mdot
.02619 .08241 Mdot
.05317 .13897 Mdot
.09696 .21823 Mdot
.20161 .3581 Mdot
.30824 .44981 Mdot
.43458 .51686 Mdot
.57015 .55857 Mdot
.02381 .07711 Mdot
.05068 .1342 Mdot
.09397 .21424 Mdot
.19609 .35554 Mdot
.29981 .44815 Mdot
.42222 .51591 Mdot
.55622 .55804 Mdot
% End of Graphics
MathPictureEnd

:[font = output; output; inactive; preserveAspect; endGroup]
Graphics["<<>>"]
;[o]
-Graphics-
:[font = subsubsection; inactive; preserveAspect; endGroup; endGroup]
This shows both modeling approaches can give good results.
:[font = section; inactive; Cclosed; dontNoPageBreakBelow; preserveAspect; startGroup]
ISSUES IN SOLUTION
:[font = subsection; inactive; noPageBreak; preserveAspect]
Once students catch on to the idea that we can plot Delta S vs. S or Delta S/Delta t vs. S to get a relationship between Delta S/ Delta t and S and thus obtain a differential equation in S they know an initial condition and they can use separation of variable techniques to solve the differential equation.  They use the solution to check against their data and to look at long term behavior of the growth model.
:[font = subsection; inactive; noPageBreak; preserveAspect]
The issue is how to prompt students to plot differences Delta S/ Delta t and S without telling them.  They will quite naturally difference the data given the juxtaposition of the columns in the table, but they need to be nudged beyond this.  The frustration of not being able to plot against time will cause some to freeze, but in groups of three or so there are usually some individual members of the several groups who will  attempt plotting the differences and then the class is off and running.  It is worth taking the time to point out how the group overcame the stumbling block and to ask those who thought of the idea what made them think that way.
:[font = subsection; inactive; noPageBreak; preserveAspect]
We have gotten students, working in groups,  to discover the essence of this approach within a 50 minute period, even doing the linear fitting by hand on graph paper, but in other situations with CAS in computer lab environment.
:[font = subsection; inactive; noPageBreak; preserveAspect; endGroup; endGroup]
An alternative solution came up when students started to attempt to predict the next size of the slick from difference equation approach.  The approach  divides the average growth rate into the actual growth  between the later observation and the next initial observation  to get an estimate of the time between  these two observations and uses this to push time until the next initial observations.  We need to modify this approach as it proves inaccurate compared to the first model solution we obtained so we take the average of the growth rates from consecutive intervals (not just over one interval, the front interval) and determine the time it takes between observation intervals. 

^*)