Chapter 6.2, Problem 19E

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 19E

The initial amount $\boxed{a=575}$ , rate of decay $\boxed{r=60\%}$ and the value of function when $t=3$ is $\boxed{y=36.8}$ rounded to nearest tenth.

Explanation of Solution

Given information:Exponential function $y=575{\left(1-0.6\right)}^t$

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 $y=575{\left(1-0.6\right)}^t$ to the exponential decay function $A=A_0{\left(1-r\right)}^t$ to identify the initial amount, and rate of decay:

  $A_0=575$ ; $r=0.6$

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

  $\begin{array}{l}r=0.6\times 100\%\\ =60\%\end{array}$

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

  $\begin{array}{l}y=575{\left(0.4\right)}^3\\ =36.8\end{array}$