Question

1. a vehicle depreciates at a rate of 22% each year after it is purchased for $30,000.

c. write an exponential equation for the value of the vehicle.
d. how much will the vehicle be worth in 8 years.

Explanation:

Step1: Define depreciation model

The standard exponential depreciation formula is $V(t) = V_0(1-r)^t$, where $V_0$ is initial value, $r$ is annual depreciation rate, $t$ is time in years.

Step2: Plug in given values (part a)

Substitute $V_0=30000$, $r=0.22$:
$V(t) = 30000(1-0.22)^t = 30000(0.78)^t$

Step3: Calculate value at t=8 (part b)

Substitute $t=8$ into the equation:
$V(8) = 30000(0.78)^8$
First compute $0.78^8 \approx 0.13701$
Then $V(8) \approx 30000 \times 0.13701$

Answer:

a. $V(t) = 30000(0.78)^t$
b. $\approx \$4110.30$