Chapter 6.2, Problem 25E

To determine

To find:the initial amount, the rate of decay of the exponential function & evaluate the function for given t value.

Answer to Problem 25E

The initial amount $\boxed{a=1}$ , rate of decay $\boxed{r=12.5\%}$ and the value of function when $t=3$ is $\boxed{y=0.7}$ rounded to nearest tenth.

Explanation of Solution

Given information:Exponential function $y={\left(\frac{7}{8}\right)}^t$

Concept Involved:

Exponential decay function is given by $A=A_0{\left(1-r\right)}^t$ where $A_0$ is the initial amount, r is the rate of decay and t is the time.

Calculation:

Rewrite the given function $y={\left(\frac{7}{8}\right)}^t$

  $\begin{array}{l}y={\left(1-\frac{1}{8}\right)}^t\\ ={\left(1-0.125\right)}^t\end{array}$

Compare the given function $y=1{\left(1-0.125\right)}^t$ to the exponential decay function $A=A_0{\left(1-r\right)}^t$ to identify the initial amount, and rate of decay:

  $A_0=1$ ; $r=0.125$

To convert the decimal to percent, multiply 100 and introduce % symbol

  $\begin{array}{l}r=0.125\times 100\%\\ =12.5\%\end{array}$

Substitute 3 for t and evaluate the value at $t=3$

  $\begin{array}{l}y={\left(\frac{7}{8}\right)}^3\\ =0.669921875\end{array}$