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

Answer:

D

Step-by-step 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:

$y=\frac{1}{2}A$

Substitute this for y:

$\frac{1}{2}A=A(1-r)^t$

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

$\frac{1}{2}A=A(1-0.1)^t$

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

$\frac{1}{2}=(1-0.1)^t$

Subtract within the parentheses:

$\frac{1}{2}=(0.9)^t$

Take the natural log of both sides:

$\ln(\frac{1}{2})=\ln((0.9)^t})$

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

$\ln(\frac{1}{2})=t\ln(0.9)$

Divide both sides by ln(0.9):

$t=\frac{\ln(\frac{1}{2})}{\ln(0.9)}$

Use a calculator. So, the car is approximately:

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

Our answer is D.

And we're done!