Question

Rich is comparing the cost of maintaining his car with the depreciation value of the car.

The value starts at $20,000 and decreases by 15% each year.
The maintanance cost is $500 the first year and increases by 28% per year.
When will the maintenance cost and the value be the same.
Please explain step by step
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Answer 1

The maintance cost and the value will be the same in 9 years.

Explanation:

Exponential function:

An exponential function has the following format:

`y(t) = y(0)(1 + r)^t`

In which r is the rate of change.

The value starts at $20,000 and decreases by 15% each year.

This means that
`V(0) = 20000, r = -0.15`

So

`V(t) = V(0)(1 + r)^t`

`V(t) = 20000(1 - 0.15)^t`

`V(t) = 20000(0.85)^t`

The maintance cost is $500 the first year and increases by 28% per year.

This means that
`M(0) = 500, r = 0.28`. So

`M(t) = M(0)(1 + r)^t`

`M(t) = 500(1 + 0.28)^t`

`M(t) = 500(1.28)^t`

When will the maintenance cost and the value be the same.

This is t for which:

`V(t) = M(t)`

So

`20000(0.85)^t = 500(1.28)^t`

`((1.28)^t)/((0.85)^t) = (20000)/(500)`

`((1.28)/(0.85))^t = 40`

`\log{((1.28)/(0.85))^t} = \log{40}`

`t\log{((1.28)/(0.85))} = \log{40}`

`t = \frac{\log{40}}{\log{((1.28)/(0.85))}}`

`t = 9`

The maintance cost and the value will be the same in 9 years.