Chapter 6.4, Problem 20E

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 20E

The initial amount $\boxed{a=8}$ , rate of decay $\boxed{r=15\%}$ and the value of function when $t=3$ is $\boxed{y\approx 4.9}$ rounded to nearest tenth.

Explanation of Solution

Given information : Exponential function $y=8{\left(1-0.15\right)}^t$

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:

Compare the given function $y=8{\left(1-0.15\right)}^t$ to the exponential decay function $A=A_0{\left(1-r\right)}^t$ to identify the initial amount, and rate of decay:

  $A_0=8$ ; $r=0.15$

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

  $\begin{array}{l}r=0.15\times 100\%\\ =15\%\end{array}$

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

  $\begin{array}{l}y=8{\left(0.85\right)}^3\\ =4.913\end{array}$