a. The value of the car decreases by 12% each year. This is an example of exponential decay. The general formula for exponential decay is:

$a_n = a_0 * (1 - r)^n$

where:
- $a_n$ is the value at the start of the nth year
- $a_0$ is the initial value
- r is the rate of decrease (in decimal form)
- n is the number of years

In this case, the initial value $a_0$ is $32,000, the rate of decrease r is 12% or 0.12, and n is the number of years. So the rule for the value of the car at the start of the nth year is:

$a_n = 32000 * (1 - 0.12)^n$

b. To find the value of the car after 7 years, we substitute n = 7 into the formula:

$a_7 = 32000 * (1 - 0.12)^7$

Calculating this gives:

$a_7 = 32000 * (0.88)^7 ≈ $13,764.79$

So, the value of the car after 7 years is approximately $13,765 when rounded to the nearest dollar.

Question

a. The value of the car decreases by 12% each year. This is an example of exponential decay. The general formula for exponential decay is:

$a_n = a_0 * (1 - r)^n$

where:
- $a_n$ is the value at the start of the nth year
- $a_0$ is the initial value
- r is the rate of decrease (in decimal form)
- n is the number of years

In this case, the initial value $a_0$ is $32,000, the rate of decrease r is 12% or 0.12, and n is the number of years. So the rule for the value of the car at the start of the nth year is:

$a_n = 32000 * (1 - 0.12)^n$

b. To find the value of the car after 7 years, we substitute n = 7 into the formula:

$a_7 = 32000 * (1 - 0.12)^7$

Calculating this gives:

$a_7 = 32000 * (0.88)^7 ≈ $13,764.79$

So, the value of the car after 7 years is approximately $13,765 when rounded to the nearest dollar.

Knowee AI · Accepted Answer