Question

For each one-year period after a car was purchased, its value at the end of the year was 17% less than its value at the beginning of the year. State whether the value of the car as a function of time after it was purchased is best modeled with a linear function, a quadratic function, or an exponential function and explain why. Enter your answer and your explanation in the space provided. If the value of the car 2 years after it was purchased is $17,842.51, what was the value of the car when it was purchased?

Answer 1

The value of the car as a function of time after it was purchased is best modeled with an exponential function.

The Value of car when it is purchased = $25,900

Explanation:

As given,

The value at the end of the year was 17% less than its value at the beginning of the year.

⇒ value at the end of the year = (100% - 17% ) = 83% of value at the beginning of the year.

Now,

Let the Value of car when it is purchased = x

So,

Price at the end of 1 year = 83% of x = 0.83x

Price at the end of 2 year = 83% of 0.83 x = (0.83)² x

Price at the end of 2 year = 83% of (0.83)² x = (0.83)³ x

and so on

Price at the end of n years = (0.83)ⁿ x

∴ we get

The value of the car as a function of time after it was purchased is best modeled with an exponential function.

Now,

Value of car after 2 year = (0.83)² x = $17,842.51

⇒0.6889x = $17,842.51

⇒x =
`(17,842.51)/(0.6889)` = $25,900

∴ we get

The Value of car when it is purchased = x = $25,900