Question

An equation for the depreciation of a car is given by y = A(1 – r)t , where y = current value of the car, A = original cost, r = rate of depreciation, and t = time, in years. The value of a car is half what it originally cost. The rate of depreciation is 10%. Approximately how old is the car?

3.3 years
5.0 years
5.6 years
6.6 years

Answer 1

D; 6.6 years.

Explanation:

We know that the equation for the depreciation of the car is:

`y=A(1-r)^t`

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:

`\displaystyle y=(1)/(2)A`

Substitute this for y:

`\displaystyle (1)/(2)A=A(1-r)^t`

We also know that the rate of depreciation is 10% or 0.1. Substitute 0.1 for r:

`\displaystyle (1)/(2)A=A(1-0.1)^t`

So, let's solve for t. Divide both sides by A:

`\displaystyle (1)/(2)=(1-0.1)^t`

Subtract within the parentheses:

`\displaystyle (1)/(2)=(0.9)^t`

Take the natural log of both sides:

`\displaystyle \ln\left((1)/(2)\right)=\ln((0.9)^t})`

Using the properties of logarithms, we can move the t to the front:

`\displaystyle \ln\left((1)/(2)\right)=t\ln(0.9)`

Divide both sides by ln(0.9):

`\displaystyle t=(\ln(0.5))/(\ln(0.9))`

Use a calculator. So, the car is approximately:

`t\approx6.6\text{ years old}`

Our answer is D.

And we're done!