Question

ppose that the dollar value v(t) of a certain car that is t years old is v(t)=32,000(1.24)^t find the initial value of the car. $32000 does the function represent growth or decay? growth decay by what percent does the value of the car change each year? %

Explanation:

Step1: Recall the exponential function form

The general form of an exponential function is \( v(t) = a(b)^t \), where \( a \) is the initial value and \( b \) determines growth or decay. If \( b>1 \), it's growth; if \( 0 < b < 1 \), it's decay. The growth rate \( r \) is given by \( b=1 + r \), so \( r=b - 1 \).

Step2: Identify \( b \) from the given function

In the function \( v(t)=32000(1.24)^t \), we have \( b = 1.24 \).

Step3: Calculate the growth rate

To find the percentage change, we use \( r=b - 1 \). Substituting \( b = 1.24 \), we get \( r=1.24 - 1=0.24 \). To convert this to a percentage, we multiply by 100: \( 0.24\times100 = 24\% \).

Answer:

24%