Question

Amy bought a car in 2009 valued at $32,500. The car is expected to depreciate at a rateof 11.1% annually. In how many years will Amy's vehicle be worth 50% of its originalvalue? Round your answer to the nearest tenth of a year,

Answer 1

5.9 years

EXPLANATION :

Exponential function can be expressed as :

`y=A(1\pm r)^t`

where A = initial amount

r = (+) growth or (-) decay rate

t = time

y = amount after t years

From the problem, the initial value of the car is A = $32,500

It depreciates at a rate of 11.1% annually, so r = -11.1% or -0.111

The value of the car will be 50% of its original value, so y = 0.50(32,500) = $16,250

Using the formula above :

`\begin{gathered} 16250=32500(1-0.111)^t \\ (16250)/(32500)=(0.889)^t \\ \\ 0.5=(0.889)^t \\ \text{ Take the ln of both sides :} \\ \ln(0.5)=\ln(0.889)^t \\ \ln(0.5)=t\ln(0.889) \\ \\ t=(\ln0.5)/(\ln0.889)=5.89\sim5.9yrs \end{gathered}`