Question

Tenzin has purchased a minivan for $35 000. The value (V), in dollars, of the
minivan as a function of time (t), in years, depreciates according to the function
V(t) = 35000(0.5)^t/3. How long will it take for Tenzin's minivan to depreciate to 15% of its initial value?
(Equation included below for clarity).
$v(t) = 35000( \frac{1}{2} )^{ \frac{t}{3} }$
​

Answer 1

Answer:

About 8.2 years.

Step-by-step explanation:

The minivan was purchased for $35,000 and it depreciates according to the function:

$\displaystyle V(t)=35000\Big(\frac{1}{2}\Big)^{t/3}$

Where t is the time in years.

And we want to determine how long it will take for the minivan to depreciate to 15% of its initial value.

First, find 15% of the initial value. This will be:

$0.15(35000)=5250$

Therefore:

$\displaystyle 5250=35000\Big(\frac{1}{2}\Big)^{t/3}$

Solve for t. Divide both sides by 35000:

$\displaystyle 0.15=\Big(\frac{1}{2}\Big)^{t/3}$

We can take the natural log of both sides:

$\displaystyle \ln(0.15)=\ln(0.5^{t/3})$

Using logarithmic properties:

$\displaystyle \ln(0.15)=\frac{t}{3}\ln(0.5)$

Therefore:

$\displaystyle t=\frac{3\ln(0.15)}{\ln(0.5)}=8.2108...$

So, it will take about 8.2 years for Tenzin's minivan to depreciate to 15% of its initival value.