Question

A new car is purchased for $41,000 and over time its value depreciates by one half every 4 years. What is the value of the car 6 years after it was purchased, to the meanest hundred dollars?

Answer 1

The general exponential decay formula is:

`\begin{gathered} y=a\cdot b^x \\ \text{with b < 1} \end{gathered}`

In this case, y represents the value of the car and x represents time. At x = 0, y = 41000 (initial value), then:

`\begin{gathered} 41000=a\cdot b^0^{} \\ 41000=a \end{gathered}`

At x = 4, y = 41000/2, then:

`\begin{gathered} (41000)/(2)=41000\cdot b^4 \\ (1)/(2)=b^4 \\ \log _(10)((1)/(2))=4\log _(10)b \\ -(0.301)/(4)=\log _(10)b \\ 10^(-0.07525)=b \\ 0.841=b \end{gathered}`

At x = 6,

`\begin{gathered} y=41000\cdot0.841^6 \\ y=41000\cdot0.3538 \\ y=14500 \end{gathered}`

The value of the car is $14500 after 6 years