Question

question #1
steven purchased a new motorcycle. . the equation $v(t) = 12,000(0.75)^t$ represents the value
of a motorcycle $t$ years after it was purchased.
(a) what was the initial price of the
motorcycle?
(b) the motorcycles value is depreciating at
what rate each year?
calculate the value of the car after 5 years:

Explanation:

Step1: Find initial price (t=0)

Substitute $t=0$ into $V(t)=12,000(0.75)^t$.
$V(0)=12,000(0.75)^0=12,000\times1=12,000$

Step2: Find annual depreciation rate

The decay factor is $0.75$, so depreciation rate = $1 - 0.75$.
$\text{Rate}=1 - 0.75=0.25=25\%$

Step3: Calculate value after 5 years

Substitute $t=5$ into $V(t)=12,000(0.75)^t$.
$V(5)=12,000(0.75)^5=12,000\times0.2373046875=2847.65625$

Answer:

(a) $\$12,000$
(b) $25\%$ per year
Value after 5 years: $\$2847.66$ (rounded to nearest cent, or $\$2847.65$ if unrounded)