Exponential Modelling Question

Question

In a study of the change in an insect population, there were about 170 insects four weeks after the study began and about 320 insects in week 6.  Round your decimal answers to the nearest hundredth.

What is the exponential model:   f(x) = answer(answer)^x

Estimate the initial number of insects: Answer

How many insects would you predict for week 2: Answer

Warning: You are to do this question by hand. If you use a statistical utility, your answers will be off because the statistical utility uses more than 2 decimal places.

Answer 1

Hello,

This question is flawed at it's foundation because you would almost never use exponential modeling for a data set with only two points. In addition, i'm not too sure how you could, would, or should quantify a number of objects (for instance, insects) to a decimal place as you either have one or you dont.

Now to perform this by hand, you would first divide the later measurement by the earlier measurement to obtain a growth ratio.

320/170 = 1.88

Thus, you have a growth factor of 1.88 over the course of 2 time intervals (weeks). To find a weekly rate, you need to determine the exponent to apply to the number 1.88 in order to "shrink" your ratio (and create an exponential function from a single number) in order to apply on a weekly basis.

1.88 ^(1/2) =1.37

Now we need to rearrange this information into the standard form: E(x) = a * b^t

Since we know at 4 weeks, we had 170 insects we start with:

170 = a * 1.37 ⁴

a = 170/(3.52)

a = 48.30 **This is the answer to part 2:

Our final exponential model results as:

E(x) = 48.3 * 1.37 ^t where t is the number of weeks *This is the answer to part 1

Now we can solve part 3: Estimate the population at 2 weeks. To do this, just plug in your new value for "t"

E(x) = 48.3 * 1.37 ² = 90.65 ***This is the answer to part 3