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.
