(*^ ::[ 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, L0, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1, 12, "Times"; ; fontset = leftfooter, L0, 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; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] ANTDIG :[font = section; inactive; preserveAspect; startGroup] BRIEF ABSTRACT :[font = subsection; inactive; preserveAspect; endGroup] We employ the derivative and antiderivative to model the time it takes an ant to dig a tunnel of length x. This is a paper and pencil activity and does not need technology. :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsection; inactive; preserveAspect; endGroup] FileName: ANTDIG Full title: Model for the time it takes an ant to build a tunnel Last Revision Date: 27 May 1996. Developer: Brian J. Winkel, Department of Mathematical Sciences, United States Military Academy, West Point NY 10996 USA. Phone: 914-938-3200. Email: ab3646@usma2.usma.edu. FAX: 914-938-2409. Contact: Brian J. Winkel, Department of Mathematical Sciences, United States Military Academy, West Point NY 10996 USA. Phone: 914-938-3200. Email: ab3646@usma2.usma.edu. FAX: 914-938-2409. Aaron D. Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8151. Email: Aaron.Klebanoff@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] Create a mathematical model to help answer the question: :[font = subsection; inactive; preserveAspect; startGroup] (1) How long does it take an ant to build a tunnel? That is, construct a mathematical model to find T(x) the time it will take an ant to build a tunnel x units long. :[font = subsubsection; inactive; preserveAspect; endGroup] More precisely: How long does it take an ant to dig a straight tunnel of length x in uniform, moist, fine sand into the side of a vertical wall? :[font = subsection; inactive; preserveAspect] Use your model to answer the following question: :[font = subsection; inactive; preserveAspect] (2) If we double the length of the tunnel then what happens to the time it takes to dig the tunnel? :[font = subsection; inactive; preserveAspect] From observations about ants and a desire to emphasize the simplicity of the situation rather than the complexity involved one can safely make the following five assumptions: :[font = subsection; inactive; preserveAspect] (a) The ant will build a level straight tunnel. (b) The ant will build in uniform, moist, fine sand. (c) The ant will walk as fast when it is carrying something as when it is unburdened. It would take more energy to do so, but the pace is the same. (d) The tunnel's cross-sectional area is constant. (e) The ant is digging into the side of a sand wall and the entry hole is high above the ground at the foot of the wall. :[font = subsection; inactive; preserveAspect; endGroup] This last assumption we throw in so that the issue of what to do with thesand when we get it to the mouth of the tunnel does not complicate the modeling activity too much. :[font = section; inactive; Cclosed; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; preserveAspect; endGroup] Derivative, anti-derivative, linear and quadraitc function, initial conditions, modeling. :[font = section; inactive; Cclosed; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; preserveAspect; startGroup] ISSUES RELATED TO THE PROBLEM :[font = subsubsection; inactive; preserveAspect] Our solution is not so much a solution as a set of leads the teacher may use to get the class moving and keep them going in a group modeling mode. :[font = subsubsection; inactive; preserveAspect] The process of mathematical modeling consists of a number of steps. These include accepting a situation, making some assumptions, converting the assumptions into mathematical statements, solving the mathematics, and interpreting the results in light of the situation. :[font = subsubsection; inactive; preserveAspect] We discuss a modeling activity which has proven to be of interest to regular college level calculus classes as well as high school groups. We have used it at various times and for a number of different purposes: to motivate the definition of the derivative in practice, to show the modeling process and to get students involved in that process, to confirm the notion of the antiderivative, and to permit students to see that mathematics can be applied to other areas of interest, in this case, entomology. :[font = subsubsection; inactive; preserveAspect; endGroup] Students in civil engineering are intrigued with time it takes to build a tunnel and can appreciate the analysis and the conclusions drawn by this model for the time it takes to build a tunnel, especially with regard to the time it takes to build a tunnel of length 2x as compared to the time it takes to build a tunnel of length x. :[font = subsection; inactive; preserveAspect; startGroup] Prerequisites :[font = subsubsection; inactive; preserveAspect; endGroup] Definition of derivative, notion of antiderivative. :[font = subsection; inactive; preserveAspect; startGroup] Time allotment - time management :[font = subsubsection; inactive; preserveAspect; endGroup] We have finished this problem with the class participating in one 50 minute period. :[font = subsection; inactive; preserveAspect; startGroup] Expectations :[font = subsubsection; inactive; preserveAspect; endGroup] Students in course evaluations point to this exercise as useful, for it shows them the modeling process and it shows how their mathematics can be used. Since we only use this as an interjection, we do not build upon this model, but rather return to the course material. :[font = subsection; inactive; preserveAspect; startGroup] Future payoffs :[font = subsubsection; inactive; preserveAspect; endGroup] The activity of listing a large list of assumptions and concerns and then reducing to the small list necessary to formulate a mathematical model is worth practicing early and repeatedly. :[font = subsection; inactive; preserveAspect; startGroup] Extensions :[font = subsubsection; inactive; preserveAspect; endGroup] One could ease up the assumptions, e.g. the ant travels more slowly, say 75% as fast when carrying material or the tunnel is built at an angle causing down to be easer than up. Ask how much longer to build a tunnel which is n times as long. :[font = subsection; inactive; preserveAspect; startGroup] References and Sources :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] Ants, Tunnels, and Calculus: An Exercise in Mathematical Modeling, Brian J. Winkel, The Mathematics Teacher, April 1994, 87(4): 284-287. :[font = section; inactive; Cclosed; preserveAspect; startGroup] POSSIBLE SOLUTION(S) :[font = subsection; inactive; preserveAspect] First, identify more expressions, having identified the length as x in our revision of the original question. We let T(x) be the time necessary to build a straight tunnel of length x in uniform, moist, fine sand into the side of a vertical wall. And we now know that we are seeking a precise description for T(x) . :[font = subsection; inactive; preserveAspect] At this point several students may make conjectures as to the nature of T(x) . ``I know. It is some constant times x .'' ``Why?'' is our response. It sure sounds good - the longer the tunnel the more time it takes . We all make a note of this reasonable observation as a characteristic which any solution for T(x) must possess. :[font = subsection; inactive; preserveAspect] We do not offer any reasons why the linear model (constant times x ) may not be appropriate. We do point out that stating the model is linear without some support may not be ``the solution'' to determining just how long it takes an ant to build a tunnel of length x . :[font = subsection; inactive; preserveAspect] Moreover, we ask, ``Is it reasonable that if we double the tunnel length the time to construct the longer tunnel is merely twice as long? After all, isn't this what this linear model would predict? Does this linear model take into account the fact that the ant will have to drag material from the deeper second half of the tunnel all the way through the first half?'' We point out that perhaps we had better take a closer look at the situation. :[font = subsection; inactive; preserveAspect] Perhaps from knowing the time it takes to build a small section of the tunnel we can build up to a larger section of the tunnel from the assumptions more directly rather than just jumping on a function offered ``by the Class,'' e.g. TC(x) = k x . :[font = subsection; inactive; preserveAspect] We ask the question, ``How long does it take to dig out the section from distance x to distance x+h ?'' Quickly the students respond, ``The time it takes to dig to x+h less the time to dig to x . That's T(x+h) - T(x) .'' :[font = subsection; inactive; preserveAspect] Thus we have the time to extend the tunnel from distance x to x+h as T(x+h) - T(x) . And what might we be able to say about this time? :[font = subsection; inactive; preserveAspect] Since the ant must bring all the sand in this little x to x + h section out a distance x and since we assume that the ant travels at a constant rate - with or without baggage - we can assume that the time to cart out all the stuff from the x to x+h section is proportional to x , i.e. T(x+h) - T(x) = b x . :[font = subsection; inactive; preserveAspect] Based on our previous discussion and the proposed linear model for T(x) itself which was offered earlier we postulate that the time it takes to extend the tunnel h units (for small h ) is also proportional to h , for the longer the section, the longer the time needed to dig out the section. In fact, we make it reasonable by saying, ``If h is 10 little granules high by 1 granule deep, then the ant has to make 10 trips back out the tunnel. If h is 10 little granules high by 2 granules deep, then the ant has to make 20 trips back out the tunnel - all over the same trip distance, x .'' Thus it seems reasonable that T(x+h) - T(x) = c h . :[font = subsection; inactive; preserveAspect] Since this tunnel is into the side of the hill, the ant merely drops the sand over the edge at the tunnel opening and this does not affect the time for digging. :[font = subsection; inactive; preserveAspect] Now we point out that we have T(x+h) - T(x)= c h and we also have T(x+h) - T(x) = b x . This means c h = b x for all x and all h . The only way this can happen is if c = a x and b = a h , i.e. a is a constant of proportionality. :[font = subsection; inactive; preserveAspect] Thus, we produce the equation T(x+h) - T(x) = a x h . :[font = subsection; inactive; preserveAspect] At this point, if the class is studying the derivative, we ask if they see the makings of a derivative in this equation. In all cases we point out that our objective is to get a full description of T(x) , perhaps to solve for T(x) . :[font = subsection; inactive; preserveAspect] We again check our assumptions. Have we used them? Have we used them well? :[font = subsection; inactive; preserveAspect] We mention the students' developing skills for differentiation which demand the form (f(x+h)-f(x))/h . In this case we get the difference expression (T(x+h) - T(x))/h = a x . :[font = subsection; inactive; preserveAspect] We recall that we wish information about T(x) , not T(x) and T(x+h) , and recalling the definition of derivative if the students do not mention it (and someone usually does) we proceed to obtain T'(x) =Limit as h goes to 0 of (T(x+h) - T(x))/h = a x . :[font = subsection; inactive; preserveAspect] Thus to find T(x) we need to find a function T(x) which has a derivative equal to a x for some constant a. :[font = subsection; inactive; preserveAspect] In the classes in which the formal antiderivative or indefinite integral concept has been covered we know that the answer is T(x) = a/2 x^2 + k . :[font = subsection; inactive; preserveAspect] And in those classes in which we have not covered the indefinite integral we ask them to think of a function which has x as a derivative, then one which has a x as a derivative. They usually get the quadratic part, and upon being prompted, ``Could we add, say 4, or 5, or k to your answer?'' they will see the general form. :[font = subsection; inactive; preserveAspect] Now we are getting somewhere. And the group usually feels that way, for they have a candidate for T(x) which was built upon the simple assumptions and not just thrown out to us. But there is concern about the two constants a and k . We ask the students how they might determine them. :[font = subsection; inactive; preserveAspect] We sometimes need to prompt them with such questions as, ``How long does it take the ant to build a tunnel of length x = 0 ?'' And will this knowledge give us information about these constants? Sure enough, 0 = T(0) = a/2 (0)^2 + k = k . Thus we have a simpler model and one less constant T(x) =a/2 x^2 . :[font = subsection; inactive; preserveAspect] But what about the constant a? If someone knew some other data point, e.g., how long it takes to build a tunnel of length, say x = 1 , then we could determine T(1) = a/2(1)^2 and thus a = 2 T(1). But we do not have any more realistic, known data. :[font = subsection; inactive; preserveAspect] Thus we are comfortable with the constant a and our model (a one parameter model), for T(x), the time necessary for an ant to build a straight tunnel of length x into the side of a vertical wall. :[font = subsection; inactive; preserveAspect] We look at T(x) =a/2 x^2 and ask, ``Is this a reasonable model?'' and ``Why is it more reasonable than the `Class offered' model TC(x) = kx ?'' :[font = subsection; inactive; preserveAspect] Perhaps we can answer both questions with the same issue, namely, what if we double the length of the tunnel from x to 2x ? How does that effect the time to build the tunnel? :[font = subsection; inactive; preserveAspect] With the model TC(x) = kx , we see that doubling the length from x to 2x has the following effect: TC(2x) = k(2x) = 2(kx) = 2TC(x) . :[font = subsection; inactive; preserveAspect] Thus the class model predicts that it takes twice as long to build a tunnel of length 2x as it does to build a tunnel of length x . :[font = subsection; inactive; preserveAspect] This sounds good -- but wait. This says it takes T(x) time to build the initial tunnel of length x and it takes the same amount of time to build the next x units of the tunnel. This is unreasonable for the ant has to carry all the material from the deeper section of the tunnel through the outer section of tunnel in addition to the distance in the deeper section. So, surely the time to build say the first half of the tunnel in cannot be the same as the time to build the second half in. With this, we finally dismiss the linear model. (We could have done this earlier, even before we arrived at our answer, and, indeed, we have done this in some instances.) :[font = subsection; inactive; preserveAspect] Now let us consider our derived model, T(x) = a/2x^2 , and ask the questions, ``What if we double the length of the tunnel from x to 2x ? How does that effect the time to build the tunnel?'' T(2x) = a/2 (2x)^2 = a/2 4x^2 = 4 a/2 x^2 = 4T(x) . :[font = subsection; inactive; preserveAspect] Thus it takes four times as long to build a tunnel of length 2x as it does to build a tunnel of length x . We suggest that this IS interesting and it is reasonable now that we believe doubling the length of the tunnel from x to 2x should more than double the time it takes, T(x), to build a tunnel of length x . :[font = subsection; inactive; preserveAspect; endGroup] These discussions make the model appear reasonable as it seems to fit well with our perceptions of reality. We interpret the model as only a rough estimation, for unless we have more data, we cannot confirm or deny the model. But when we teach calculus for engineering students this analysis suggests issues for the civil engineering students who will eventually be making bids on projects - namely doubling the length of a tunnel will not merely double the cost! :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsection; inactive; preserveAspect; endGroup; endGroup] The teacher needs to guide the modeling process, but it is good to let students conjecture at the start and from time to time in the process to permit the introduction of extraneous and "wild" ideas. This serves to show off the modeling process in which we often reduce the problem to tractable mathematics. ^*)