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.

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.)

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:

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.

Here we determine the (time, size) data

and plot the revised data to compare with the function we obtained from our first approach to the problem.

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.

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.

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:

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.

Here we determine the (time, size) data

and plot the revised data to compare with the function we obtained from our first approach to the problem.

That looks better and we have a reasonable model here.