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( (1)/(2) )^{ (t)/(3) }`
​

Answer 1

About 8.2 years.

Explanation:

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

`\displaystyle V(t)=35000\Big((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((1)/(2)\Big)^(t/3)`

Solve for t. Divide both sides by 35000:

`\displaystyle 0.15=\Big((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)=(t)/(3)\ln(0.5)`

Therefore:

`\displaystyle t=(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.