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.
Solution
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.
