Question

The value of a car can be modeled by the equation y = 24,000(0.845)t where t is the number of

years since the car was purchased.

a. After how many years will the value of the car be $10,000?


b. Use the model to predict the value of the car after 50 years. Is this a reasonable value?

Explain.

Answer 1

Answer: 5.2 years, $5.28

Explanation:

Given

The value of a car is given by
`y=24,000\left(0.845\rght)^t`

Substitute
`\$10,000` for
`y`

`\Rightarrow 10,000=24,000\left(0.845\right)^t\\\\\Rightarrow \left(0.845\right)^t=(5)/(12)\\\\\Rightarrow \left(0.845\right)^t=0.41666\\\text{Taking log both sides}\\\Rightarrow t\ln (0.845)=\ln (0.4166)\\\\\Rightarrow t=(\ln 0.4167)/(\ln 0.845)\\\\\Rightarrow t=5.197\approx 5.2\ \text{years}`

(b) After 50 years, it is

`\Rightarrow y=24,000\left(0.845\rght)^(50)`

`\Rightarrow y=24,000* 0.00022\\\Rightarrow y=\$5.28`

No, this is not a reasonable value as it is more than 4000 times less.