Sure, let's break this problem down step-by-step.
1. Find the initial value of the car:
To determine the initial value of the car, we need to evaluate the function [tex]\( v(t) \)[/tex] when [tex]\( t = 0 \)[/tex].
[tex]\[
v(0) = 32,000 \times (0.76)^0
\][/tex]
Any number raised to the power of 0 is 1, so:
[tex]\[
v(0) = 32,000 \times 1 = 32,000
\][/tex]
Therefore, the initial value of the car is [tex]$32,000.
2. Determine if the function represents growth or decay:
The function \( v(t) = 32,000 \times (0.76)^t \) involves an exponential term. The base of the exponential function is 0.76, which is less than 1.
- If the base of an exponential function is greater than 1, it represents growth.
- If the base of an exponential function is less than 1, it represents decay.
Since the base (0.76) is less than 1, this function represents decay.
3. By what percent does the value of the car change each year:
To find the percentage change each year, we can use the base of the exponential function. The base 0.76 indicates that the value retains 76% of its value each year.
To find the percentage decrease, we calculate:
\[
\text{Percentage decrease} = (1 - 0.76) \times 100
\]
\[
\text{Percentage decrease} = 0.24 \times 100 = 24\%
\]
Thus, the value of the car decreases by 24% each year.
Summary:
- The initial value of the car is $[/tex]32,000.
- The function represents decay.
- The value of the car decreases by 24% each year.
