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?

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 [tex]y[/tex] is the current value and [tex]A[/tex] is the original cost. It means that [tex]y[/tex] is half of [tex]A[/tex]: [tex]y=\frac{A}{2}[/tex].

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

For finding the value of [tex]t[/tex], you must replace the values of [tex]y[/tex] and [tex]r[/tex] in the depreciation formula:

[tex]y=A\cdot (1-r)^t[/tex]

[tex]\frac{A}{2}=A\cdot (1-0.1)^t[/tex]

After cancelling the variable [tex]A[/tex] the equation would be:

[tex]\frac{1}{2}=(0.9)^t[/tex]

For finding the value of [tex]t[/tex] you must apply natural algorithm in both sides:

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

[tex]ln(0.5)=t\cdot ln(0.9)[/tex]

[tex]t=\frac{ln(0.5)}{ln(0.9)}[/tex]

[tex]t=6.5788[/tex]

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

[tex]months=12\times 0.5788=6.9456[/tex]

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