Question

a car was purchased for $26,500. each year since, the resale value has decreased by 21%. let t be the number of years since the purchase. let y be the resale value of the car, in dollars. write an exponential function showing the relationship between y and t.

Explanation:

Step1: Recall 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 decrease (in decimal), and \( t \) is time.

Step2: Identify \( a \) and \( r \)

Here, the initial value \( a = 26500 \) (the purchase price), and the rate of decrease \( r = 0.21 \) (since 21% = 0.21).

Step3: Substitute into the formula

Substitute \( a = 26500 \) and \( r = 0.21 \) into \( y = a(1 - r)^t \). So \( 1 - r = 1 - 0.21 = 0.79 \). Thus, the function is \( y = 26500(0.79)^t \).

Answer:

\( y = 26500(0.79)^t \)