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.

We enter the Initial Observation Data.

We enter the ten minute Later Observation Data.

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.

We get ready to plot the difference vs. IO data.

We see the difference (i.e. the derivative ~ (IO(t+10) - IO(t))/10) is a linear function of the initial size IO(t).

And we determine the linear relationship from a best fit command in Mathematica.

We plot the line and then the data and the line and see that the fit looks quite good.

Thus we have an approximation to a differential equation in the function OilSize[t] which we proceed to solve.

We solve and pick off the solution to the differential equation.

We plot the solution function called OilSlick[t].

Now we determine the times at which our OilSlick[t] function is at the IO times.

And we list these times, t, and the corresponding OilSlick[t] values.

with a plot to follow of the IO data over the function OilSLick[t].

Now we determine the times at which our OilSlick[t] function is at the IO(t + 10) times.

And we list these times, t, and the corresponding OilSlick[t] values.

with a plot to follow of the IO(t+10) data over the function OilSLick[t].

The following plot shows that the initial and 10 minute later data fit the model very well.

And this zoom plot confirms the fit even more.

Finally here is what the OilSlick model predicts the 10 minutes later size of the slick should be:

We compare this with the 10 minutes later data,

even differencing them at each increment to see how close our model is.