Question

How many years would it be when the value of the boat is $40,000?

Answer 1

It will take 13 years.

Step-by-step explanation

First, we have to write the equation that represents the situation.

It is a depreciation, which means that it is a compound decrease.

Compound decrease is generally given as:

`\begin{gathered} A=P(1-r)^t \\ A=\text{amount;} \\ P=\text{initial amount} \\ r=\text{rate} \\ t=\text{amount of time} \end{gathered}`

Therefore, from the question:

`\begin{gathered} A=95000(1-(6.25)/(100))^t=90000(1-0.0625)^t \\ A=95000(0.9375)^t \end{gathered}`

To find the number of years after which the boat will have a value of $40,000, we have to find t when A is $40,000.

That is:

`\begin{gathered} 40000=95000(0.9375)^t \\ \text{Divide both sides by 95000:} \\ (40000)/(95000)=0.9375^t \\ 0.42=0.9375^t \\ Find\text{ the log of both sides:} \\ \log (0.42)=\log (0.9375)^t=t\cdot\log (0.9375) \\ D\text{ivide both sides by log(0.9375)}\colon \\ t=(\log(0.42))/(\log(0.9375)) \\ t\approx13\text{ years} \end{gathered}`

It will babout 13 years for the value for the value of the car to be $40,000.