Chapter 6.2, Problem 24E

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 24E

The initial amount $\boxed{a=1250}$ , rate of decay $\boxed{r=13.5\%}$ and the value of function when $t=3$ is $\boxed{h\left(t\right)\approx 809.0}$ rounded to nearest tenth.

Explanation of Solution

Given information:Exponential function $h\left(t\right)=1250{\left(0.865\right)}^t$

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 $h\left(t\right)=1250{\left(0.865\right)}^t$

  $h\left(t\right)=1250{\left(1-0.135\right)}^t$

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

  $A_0=1250$ ; $r=0.135$

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

  $\begin{array}{l}r=0.135\times 100\%\\ =13.5\%\end{array}$

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

  $\begin{array}{l}h\left(t\right)=1250{\left(0.865\right)}^3\\ =809.01828125\end{array}$