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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) }
​

1 Answer

1 vote

Answer:

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.

User KennyZ
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