Chapter 6.4, Problem 9E

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 9E

The value of initial amount $a$ is $1500$ , rate of growth $r$ as a percentage is $7.4\%$ and $y\approx 2143.4$at$t=5$ .

Explanation of Solution

Given information:

The exponential functionis $f\left(t\right)=1500{\left(1.074\right)}^t$ .

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 $f\left(t\right)=1500{\left(1.074\right)}^t$ .

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

  $a=1500$ and $1+r=1.074$

Calculate the value of $r$ rate of growth.

  $\begin{array}{c}1+r=1.074\\ r=1.074-1\\ r=0.074\end{array}$

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

  $\begin{array}{c}r\%=0.074\left(100\right)\\ =7.4\%\end{array}$

Now, solve the given exponential function $f\left(t\right)=1500{\left(1.074\right)}^t$ at $t=5$ .

  $\begin{array}{c}f\left(t\right)=1500{\left(1.074\right)}^t\\ =25{\left(1.074\right)}^5\\ =25\left(1.4289643919\right)\\ \approx 2143.4\end{array}$

Therefore, the value of initial amount $a$ is $1500$ , rate of growth $r$ as a percentage is $7.4\%$ and $y\approx 2143.4$ at $t=5$ .

Conclusion:

Thus, the value of initial amount $a$ is $1500$ , rate of growth $r$ as a percentage is $7.4\%$ and $y\approx 2143.4$at$t=5$ .