Question

the price p, in dollars, of a specific car that is x years old is modeled by the following function. p(x)=45295·0.90^x (a) how much should a 4 - year - old car cost? (b) how much should a 9 - year - old car cost? (c) explain the meaning of the base 0.90 in this problem. (a) a 4 - year - old car should cost approximately $29718. (round to the nearest whole number as needed.) (b) a 9 - year - old car should cost approximately $□ (round to the nearest whole number as needed.)

Explanation:

Step1: Substitute x = 9 into the function

We have the function \(p(x)=45295\times0.90^{x}\). When \(x = 9\), we calculate \(p(9)=45295\times0.90^{9}\).

Step2: Calculate \(0.90^{9}\)

\(0.90^{9}=0.9\times0.9\times\cdots\times0.9\) (9 times). Using a calculator, \(0.90^{9}\approx0.38742\).

Step3: Calculate \(p(9)\)

\(p(9)=45295\times0.38742\approx17544\)

Step4: Explain the base 0.90

The base 0.90 represents the decay - factor of the car's value each year. It means that the value of the car is 90% (or 0.90 times) of its value from the previous year. In other words, the car loses 10% of its value each year.

Answer:

(a) $29718
(b) $17544
(c) The value of the car is 90% of its value from the previous year, meaning it loses 10% of its value each year.