Chapter 6.3, Problem 4E

Reminder Round all answers to two decimal places unless otherwise indicated.

Further Analysis of Population Growth This is a continuation of Exercise 3. Our goal is to make an exponential model or the data and use it to get a more accurate estimate of the rate of change in population in 1955.

a. Use regression to obtain an exponential model for population growth. (For details on the method used here, see Section $4.4$.)

b. Use the formula you found in part a to get $\frac{dN}{dt}$ in 1955.

c. How does your answer from part b of this exercise compare with your answer from part a of exercise 3? Does this agree with your answer in part $c$ of Exercise 3?

Population Growth The following table shows the population of reindeer on an island as of the given year.

$\text{date}$	$1945$	$1950$	$1955$	$1960$
$\text{population}$	$40$	$165$	$678$	$2793$

We let $t$ be the number of years since 1945, so that $t=0$ corresponds to 1945. and we let $N=N\left(t\right)$ denote the population size.

a. Approximate $\frac{dN}{dt}$ for 1955 using the average rate of change from 1955 to 1960, and explain what this number means in practical terms.

b. Use your work from part $\text{a}$ to estimate the population in 1957.

c. The number you calculated in part $\text{a}$ is an approximation to the actual rate of change. As you will be asked to show in the next exercise, the reindeer population growth can be closely modeled by an exponential function. With this in mind, du you think your answer in part $a$ is too large or too small? Explain your reasoning.