Answer :
Sure, let's break down the solution step-by-step for each part of the question.
### Part a: Explanation of the numbers 24,500 and 0.88
- 24,500: This number represents the initial value of the car in dollars, which is the purchase price when it was first bought.
- 0.88: This number represents the annual depreciation factor of the car. In this context, it means that each year the car retains 88% of its value from the previous year. This is a percentage decrease, indicating that the car loses 12% of its value every year.
### Part b: Understanding [tex]\( f(9) \)[/tex]
- [tex]\( f(9) \)[/tex]: This expression represents the value of the car 9 years after it was purchased. Plugging into the function, we calculate the car's value as follows:
[tex]\[
f(9) = 24,500 \times (0.88)^9
\][/tex]
Calculating this gives the car's value 9 years after purchase, which is approximately [tex]$7,753.72.
### Part c: Values for the graph
To plot the graph, we need to find the car's value at specific times.
- \( f(0) \): This is the starting value when the car was just purchased.
\[
f(0) = 24,500 \times (0.88)^0 = 24,500
\]
Since anything to the power of 0 is 1, the value remains as $[/tex]24,500.
- [tex]\( f(1) \)[/tex]: This is the value of the car one year after the purchase.
[tex]\[
f(1) = 24,500 \times (0.88)^1 = 21,560
\][/tex]
- [tex]\( f(2) \)[/tex]: This is the value of the car two years after the purchase.
[tex]\[
f(2) = 24,500 \times (0.88)^2 = 18,972.80
\][/tex]
These calculations give us the points [tex]\( (0, 24,500) \)[/tex], [tex]\( (1, 21,560) \)[/tex], and [tex]\( (2, 18,972.80) \)[/tex], which can be plotted on a graph to visualize the depreciation of the car's value over time. The graph will show a downward curve, reflecting how the car loses value each year.
### Part a: Explanation of the numbers 24,500 and 0.88
- 24,500: This number represents the initial value of the car in dollars, which is the purchase price when it was first bought.
- 0.88: This number represents the annual depreciation factor of the car. In this context, it means that each year the car retains 88% of its value from the previous year. This is a percentage decrease, indicating that the car loses 12% of its value every year.
### Part b: Understanding [tex]\( f(9) \)[/tex]
- [tex]\( f(9) \)[/tex]: This expression represents the value of the car 9 years after it was purchased. Plugging into the function, we calculate the car's value as follows:
[tex]\[
f(9) = 24,500 \times (0.88)^9
\][/tex]
Calculating this gives the car's value 9 years after purchase, which is approximately [tex]$7,753.72.
### Part c: Values for the graph
To plot the graph, we need to find the car's value at specific times.
- \( f(0) \): This is the starting value when the car was just purchased.
\[
f(0) = 24,500 \times (0.88)^0 = 24,500
\]
Since anything to the power of 0 is 1, the value remains as $[/tex]24,500.
- [tex]\( f(1) \)[/tex]: This is the value of the car one year after the purchase.
[tex]\[
f(1) = 24,500 \times (0.88)^1 = 21,560
\][/tex]
- [tex]\( f(2) \)[/tex]: This is the value of the car two years after the purchase.
[tex]\[
f(2) = 24,500 \times (0.88)^2 = 18,972.80
\][/tex]
These calculations give us the points [tex]\( (0, 24,500) \)[/tex], [tex]\( (1, 21,560) \)[/tex], and [tex]\( (2, 18,972.80) \)[/tex], which can be plotted on a graph to visualize the depreciation of the car's value over time. The graph will show a downward curve, reflecting how the car loses value each year.
