Question

a)a science class is observing how bacteria on a piece of bread behaves. the function f(x)=500(1.2)^x models the number of bacteria after x hours. determine and interpret the percent rate of change.
b)a person is trying to determine how much their car will be worth throughout the life of their auto loan. the function g(x)=28000(0.85)^x models the value of a car after x years. determine and interpret the percent rate of change.

Explanation:

Step1: Recall the general form of an exponential function

The general form of an exponential function is $y = a(b)^x$, where $b$ determines the rate of change. If $b> 1$, it is growth and if $0 < b<1$, it is decay. The percent - rate of change $r$ is related to $b$ by the formula $b = 1 + r$ (for growth) or $b=1 - r$ (for decay).

Step2: Analyze the function for bacteria

For the function $f(x)=500(1.2)^x$, we have $b = 1.2$. Using the formula $b = 1 + r$, we solve for $r$:
$1.2=1 + r$
$r=1.2 - 1=0.2$. To convert to a percentage, we multiply by 100. So the percent rate of change is $20\%$. This means the number of bacteria is increasing by $20\%$ per hour.

Step3: Analyze the function for car value

For the function $g(x)=28000(0.85)^x$, we have $b = 0.85$. Using the formula $b = 1 - r$, we solve for $r$:
$0.85=1 - r$
$r = 1-0.85 = 0.15$. To convert to a percentage, we multiply by 100. So the percent rate of change is $15\%$. This means the value of the car is decreasing by $15\%$ per year.

Answer:

a) The percent rate of change is $20\%$ and it represents an increase in the number of bacteria per hour.
b) The percent rate of change is $15\%$ and it represents a decrease in the value of the car per year.