Question

question 11 (multiple choice worth 5 points) (percent rate of change mc) a college student runs a side business detailing cars. in the first year, the business made $12,000. the function c(t) = 12,000 · (1.12)^t models the total yearly earnings, c(t), in dollars, for a given number of years, t. what is the percent rate of change, and how is it interpreted in the context of the problem? the percent rate of change is 88%, which means the total yearly earnings increase by 88% each year. the percent rate of change is 88%, which means the total yearly earnings decrease by 88% each year. the percent rate of change is 12%, which means the total yearly earnings increase by 12% each year. the percent rate of change is 12%, which means the total yearly earnings decrease by 12% each year

Explanation:

Step1: Analyze the exponential function form

The general form of an exponential function is \(y = a(1 + r)^t\), where \(a\) is the initial amount, \(r\) is the rate of change. In the function \(c(t)=12000(1.12)^t\), comparing with the general form, we have \(1 + r=1.12\).

Step2: Solve for \(r\)

Subtract \(1\) from both sides of the equation \(1 + r = 1.12\). So \(r=1.12 - 1=0.12\). Since \(r\) represents the rate of change, and \(r = 0.12=12\%\), and because the base of the exponential function \(1.12>1\), it means the function is increasing.

Answer:

The percent rate of change is \(12\%\), which means the total yearly earnings increase by \(12\%\) each year.