Question

question 22 (multiple choice worth 1 point)
a new car was purchased for $50,000. the function ( v(t)=50,000cdot(0.70)^t ) models the total value of the car, ( v(t) ), in dollars, after 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 30%, which means the value of the car increases by 30% each year.
the percent rate of change is 30%, which means the value of the car decreases by 30% each year.
the percent rate of change is 70%, which means the value of the car increases by 70% each year.
the percent rate of change is 70%, which means the value of the car decreases by 70% each year.

Explanation:

Step1: Analyze the exponential decay formula

The general form of an exponential decay function is \(y = a(1 - r)^t\), where \(a\) is the initial amount, \(r\) is the rate of decay, and \(t\) is the time. In the given function \(v(t)=50000(0.70)^t\), we can rewrite \(0.70\) as \(1 - 0.30\).

Step2: Interpret the rate

Comparing with the exponential decay formula \(y = a(1 - r)^t\), here \(r = 0.30\) or \(30\%\). Since it is an exponential decay function (the base \(0.70<1\)), the value of the car is decreasing.

Answer:

The percent rate of change is \(30\%\), which means the value of the car decreases by \(30\%\) each year.