Chapter 6.4, Problem 7E

To determine

To find: The initial amount $a$ and growth rate $r$ also find the value of $y$ at $t=5$ and express the answer in the nearest tenth form

Answer to Problem 7E

The value of initial amount is $a=25$ rate of growth $r$ as a percentage is $20\%$ and $y\approx 62.2$ at $t=5$ .

Explanation of Solution

Given information:

The exponential functionis $y=25{\left(1.2\right)}^t$ .

Concept used:

The exponential growth function is as follows:

  $y=a{\left(1+r\right)}^t$ .................. (1)

Here, $a$ is an initial amount, $r$ is the rate of growth and $t$ is the time interval.

Calculation:

The exponential function is $y=25{\left(1.2\right)}^t$ .

Now, compare the given exponential function with equation (1) to obtain the initial amount $a$ and rate of growth $r$ .

  $a=25$ and $1+r=1.2$

Calculate the value of $r$ rate of growth.

  $\begin{array}{c}1+r=1.2\\ r=1.2-1\\ r=0.2\end{array}$

Now, find $r$ rate of growth as a percentage.

  $\begin{array}{c}r\%=0.2\left(100\right)\\ =20\%\end{array}$

Now, solve the given exponential function $y=25{\left(1.2\right)}^t$ at $t=5$ .

  $\begin{array}{c}y=25{\left(1.2\right)}^5\\ =25{\left(1.2\right)}^5\\ =25\left(2.48832\right)\\ \approx 62.2\end{array}$

Therefore, the value of initial amount $a$ is 25, rate of growth $r$ as a percentage is $20\%$ and $y\approx 62.2$ at $t=5$ .

Conclusion:

Thus, the value of initial amount $a$ is, rate of growth $r$25as a percentage is $20\%$ and $y=62.2$ at $t=5$ .