Answer:
D; 6.6 years.
Step-by-step explanation:
We know that the equation for the depreciation of the car is:
[tex]y=A(1-r)^t[/tex]
Where y is the current cost, A is the original cost, r is the rate of depreciation, and t is the time, in years.
We are told that the value of the car now is half of what it originally cost. So:
[tex]\displaystyle y=\frac{1}{2}A[/tex]
Substitute this for y:
[tex]\displaystyle \frac{1}{2}A=A(1-r)^t[/tex]
We also know that the rate of depreciation is 10% or 0.1. Substitute 0.1 for r:
[tex]\displaystyle \frac{1}{2}A=A(1-0.1)^t[/tex]
So, let's solve for t. Divide both sides by A:
[tex]\displaystyle \frac{1}{2}=(1-0.1)^t[/tex]
Subtract within the parentheses:
[tex]\displaystyle \frac{1}{2}=(0.9)^t[/tex]
Take the natural log of both sides:
[tex]\displaystyle \ln\left(\frac{1}{2}\right)=\ln((0.9)^t})[/tex]
Using the properties of logarithms, we can move the t to the front:
[tex]\displaystyle \ln\left(\frac{1}{2}\right)=t\ln(0.9)[/tex]
Divide both sides by ln(0.9):
[tex]\displaystyle t=\frac{\ln(0.5)}{\ln(0.9)}[/tex]
Use a calculator. So, the car is approximately:
[tex]t\approx6.6\text{ years old}[/tex]
Our answer is D.
And we're done!
