Chapter ISG, Problem 7.5.5P

a.

To determine

To write: an exponential function which fits this data and can be used to predict the deer population in future years.

Answer to Problem 7.5.5P

Equation of the exponential function is $y=128{(1.25)}^x.$

Explanation of Solution

Given information: The table shows the estimated number of deer in the forest over a 3-.

year period .

Years of Study	0	1	2	3
Population	128	160	200	250

Formula used:

The general equation of an exponential function is,

  $y=a{b}^x$ where, a = initial value.

Calculation:

From the table data a =128 and the function passes through point (1,160), so:

  $\begin{array}{c}160=128b^1\\ 128b=160\\ b=\frac{160}{128}=\mathrm{1.25.}\end{array}$

So, equation of the exponential function will be:

  $y=128{(1.25)}^x.$

b.

To determine

To calculate: the average rate of change in population during those three years.

Answer to Problem 7.5.5P

The average rate of change in population during those three years is 40.67.

Explanation of Solution

Given information: The table shows the estimated number of deer in the forest over a 3-.

year period .

Years of Study	0	1	2	3
Population	128	160	200	250

Calculation:

From the table the rate of change in population in first year = (160-128) =32.

The rate of change in population in second year = (200-160) =40.

The rate of change in population in third year = (250-200) =50.

So, the average rate of change in population during those three years:

  $=\frac{32+40+50}{3}=\mathrm{40.67.}$

c.

To determine

To make: a conjecture.

Answer to Problem 7.5.5P

The deer population continues increase by the average rate of change 40.67 as year increases.

Explanation of Solution

Given information: If the population growth follows the model from part a, the deer

population to continue to increase by the value with in part b.

Calculation:

Since $y=128{(1.25)}^x.$ is an exponential growth function so the deer population continues increase by the average rate of change 40.67 as year increases.

d.

To determine

To find: how know the function is exponential, not linear.

Answer to Problem 7.5.5P

Exponential function.

Explanation of Solution

Given information: If the population growth follows the model from part a, the deer

population to continue to increase by the value with in part b

Calculation:

Given function is exponential because population change from one year to next year increases exponentially not constant.

e.

To determine

To graph: deer population increase function and their food supply function.

Answer to Problem 7.5.5P

Red curve show the graph of $y=128{(1.25)}^x$ .

Blue curve shows the graph of $F(x)=750{(0.98)}^x$ .

Explanation of Solution

Given information: Given functions are

  $y=128{(1.25)}^x,F(x)=750{(0.98)}^x$

Calculation:

The graph of the functions $y=128{(1.25)}^x,F(x)=750{(0.98)}^x$ is given below

  

Red curve show the graph of $y=128{(1.25)}^x$ .

Blue curve shows the graph of $F(x)=750{(0.98)}^x$ .