Questions: The price p, in dollars, of a specific car that is x years old is modeled by the function below. p(x)=22,275(0.88)^x (a) How much should a 2-year-old car cost? (b) How much should a 7-year-old car cost? (c) Explain the meaning of the base 0.88 in this problem. (a) A 2-year-old car should cost approximately I. (Round to the nearest whole number as needed.)

Solution Steps

To solve the given problems, we will use the provided function \( p(x) = 22,275(0.88)^x \) to calculate the price of the car for different values of \( x \). For parts (a) and (b), we will substitute \( x = 2 \) and \( x = 7 \) into the function to find the respective prices. For part (c), we will interpret the base 0.88 as the annual depreciation rate of the car's value.

To find the price of a car that is 2 years old, we substitute \( x = 2 \) into the price function \( p(x) = 22,275(0.88)^x \):

\[ p(2) = 22,275(0.88)^2 = 22,275 \times 0.7744 \approx 17,249.76 \]

Rounding to the nearest whole number, the price of a 2-year-old car is:

\[ \boxed{p(2) = 17250} \]

Next, we calculate the price of a car that is 7 years old by substituting \( x = 7 \) into the same price function:

\[ p(7) = 22,275(0.88)^7 = 22,275 \times 0.5132 \approx 9,103.25 \]

Rounding to the nearest whole number, the price of a 7-year-old car is:

\[ \boxed{p(7) = 9103} \]

The base \( 0.88 \) in the function represents the annual depreciation rate of the car's value. Specifically, it indicates that the car retains \( 88\% \) of its value each year, meaning it loses \( 12\% \) of its value annually.

Final Answer