Chapter 6.3, Problem 11Q

a.

To determine

To find: does the function represent exponential growth or exponential decay, explain.

Answer to Problem 11Q

The function represents exponential growth because the base 4 > 1.

Explanation of Solution

Given information: The function $f\left(t\right)=5{\left(4\right)}^t$ represents the number of frogs in a pond

after t years.

Calculation:

Given the function:

  $f\left(t\right)=5{\left(4\right)}^t$ .

The function represents exponential growth because the base 4 > 1.

b.

To determine

To graph: the function and describe the domain and range.

Answer to Problem 11Q

The domain is $\left[0,\infty \right]$ .

The range is $\left[5,\infty \right]$ .

Explanation of Solution

Given information: The function $f\left(t\right)=5{\left(4\right)}^t$ represents the number of frogs in a pond

after t years.

Calculation:

Given the function:

  $f\left(t\right)=5{\left(4\right)}^t$ .

Make a table of values:

t	f(t)
0	5
1	20
2	80
3	320
4	1280
5	5120
6	20,480

Graph the function $f\left(t\right)=5{\left(4\right)}^t$ on the above table data.

  

The domain is $\left[0,\infty \right]$ .

The range is $\left[5,\infty \right]$ .

c.

To determine

To find: what is the yearly percent change.

Answer to Problem 11Q

The yearly percent change $r=300\%.$

Explanation of Solution

Given information: The function $f\left(t\right)=5{\left(4\right)}^t$ represents the number of frogs in a pond

after t years.

Calculation:

To find yearly percent change:

  $\begin{array}{l}$ f(t)=5{(1+3)}^t$r=3=300\%.\end{array}$

d.

To determine

To find: how many frogs are in the pond after $4$ years.

Answer to Problem 11Q

1280 frogs are in the pond after 4 years.

Explanation of Solution

Given information: The function $f\left(t\right)=5{\left(4\right)}^t$ represents the number of frogs in a pond

after t years.

Calculation:

Have to find f(t) for t =4:

  $f(4)=5{(4)}^4=1280$ .