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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).
[tex]v(t) = 35000( \frac{1}{2} )^{ \frac{t}{3} } [/tex]
​

Sagot :

xKelvin

Answer:

About 8.2 years.

Step-by-step explanation:

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

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

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:

[tex]0.15(35000)=5250[/tex]

Therefore:

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

Solve for t. Divide both sides by 35000:

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

We can take the natural log of both sides:

[tex]\displaystyle \ln(0.15)=\ln(0.5^{t/3})[/tex]

Using logarithmic properties:

[tex]\displaystyle \ln(0.15)=\frac{t}{3}\ln(0.5)[/tex]

Therefore:

[tex]\displaystyle t=\frac{3\ln(0.15)}{\ln(0.5)}=8.2108...[/tex]

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

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