Sure! Let's go through each part of the question step-by-step.
Given problem:
[tex]\[ p(x) = 22,255 \times (0.87)^x \][/tex]
where [tex]\( p(x) \)[/tex] is the price of the car in dollars and [tex]\( x \)[/tex] is the age of the car in years.
Part (a): How much should a 4-year-old car cost?
To find the cost of a 4-year-old car, we substitute [tex]\( x = 4 \)[/tex] into the given formula:
[tex]\[
p(4) = 22,255 \times (0.87)^4
\][/tex]
The result is:
[tex]\[
p(4) = 12,749.84
\][/tex]
Therefore, a 4-year-old car should cost approximately [tex]$12,749.84.
Part (b): How much should a 7-year-old car cost?
To find the cost of a 7-year-old car, we substitute \( x = 7 \) into the given formula:
\[
p(7) = 22,255 \times (0.87)^7
\]
The result is:
\[
p(7) = 8,395.81
\]
Therefore, a 7-year-old car should cost approximately $[/tex]8,395.81.
Part (c): Explain the meaning of the base 0.87 in this problem.
The base 0.87 in the function [tex]\( p(x) = 22,255 \times (0.87)^x \)[/tex] represents the rate of depreciation per year. Specifically, it means that each year, the car retains 87% of its value from the previous year, or equivalently, the car loses 13% (since [tex]\( 1 - 0.87 = 0.13 \)[/tex]) of its value each year.
In summary:
- A 4-year-old car costs approximately [tex]$12,749.84.
- A 7-year-old car costs approximately $[/tex]8,395.81.
- The base 0.87 means the car loses 13% of its value each year.
