The price $p$, in dollars, of a specific car that is $x$ years old is modeled by the function below.
\[
p(x)=22,265(0.89)^{x}
\]
(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 0.89 in this problem.
Answer
The base 0.89 in the function represents the rate at which the car's value depreciates each year. This means that each year, the car's value decreases by 11% of its value from the previous year.
Step 1 ：Given the function \(p(x)=22,265(0.89)^{x}\), where \(p\) is the price of the car in dollars and \(x\) is the age of the car in years.
Step 2 ：To find the price of a 4-year-old car, substitute \(x = 4\) into the function to get \(p(4)=22,265(0.89)^{4}\).
Step 3 ：Calculate the value to get approximately \$13969.56.
Step 4 ：To find the price of a 9-year-old car, substitute \(x = 9\) into the function to get \(p(9)=22,265(0.89)^{9}\).
Step 5 ：Calculate the value to get approximately \$7800.69.
Step 6 ：Final Answer: (a) The cost of a 4-year-old car should be approximately \(\boxed{13969.56}\). (b) The cost of a 9-year-old car should be approximately \(\boxed{7800.69}\).
Step 7 ：The base 0.89 in the function represents the rate at which the car's value depreciates each year. This means that each year, the car's value decreases by 11% of its value from the previous year.
