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).
