an exponential function has an initial value of 500 and a decay rate of 15%. compare the average rate of…

Explanation:

Step1: Write the exponential - decay function

The general form of an exponential - decay function is $y = a(1 - r)^x$, where $a$ is the initial value and $r$ is the decay rate. Given $a = 500$ and $r=0.15$, the function is $y = 500(0.85)^x$.

Step2: Calculate the average rate of change formula

The average rate of change of a function $y = f(x)$ over the interval $[m,n]$ is $\frac{f(n)-f(m)}{n - m}$.

Step3: Calculate the average rate of change for the interval $4\lt x\lt8$

First, find $y(4)$ and $y(8)$. $y(4)=500(0.85)^4=500\times0.52200625 = 261.003125$. $y(8)=500(0.85)^8=500\times0.2724905277 = 136.2452638$. The average rate of change over the interval $4\lt x\lt8$ is $\frac{y(8)-y(4)}{8 - 4}=\frac{136.2452638 - 261.003125}{4}=\frac{- 124.7578612}{4}\approx - 31$.

Answer:

$-31$