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?

Answer 1

Answer:

The car is 6.5788 years old.

Step-by-step explanation:

The key to solve the problem is in the following sentence: "The value of a car is half what it originally cost". Keep in mind that $y$ is the current value and $A$ is the original cost. It means that $y$ is half of $A$: $y=\frac{A}{2}$.

It is assumed that the rate of depreciation is annually and its value is $10\%$. Remember that $10\%$ equals $0.1$, so $r=0.1$.

For finding the value of $t$, you must replace the values of $y$ and $r$ in the depreciation formula:

$y=A\cdot (1-r)^t$

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

After cancelling the variable $A$ the equation would be:

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

For finding the value of $t$ you must apply natural algorithm in both sides:

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

$ln(0.5)=t\cdot ln(0.9)$

$t=\frac{ln(0.5)}{ln(0.9)}$

$t=6.5788$

The previous value can be split in two parts: $6+0.5788$. The first part refers to years and the second part can be converted to months by multiplying the total months in a year ($12$) by $0.5788$.

$months=12\times 0.5788=6.9456$

Thus, the car is 6.5788 years old (which is approximately 6 years and almost 7 months).

Answer 2

A(1 – r)t , where y = current value of the car,