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Answer :
Certainly! Let's go through each part of the question step-by-step.
### Part a:
- The number 24,500 represents the initial value of the car. This is the purchase price of the car in dollars at the beginning, or when [tex]\( t = 0 \)[/tex].
- The number 0.88 is the depreciation factor. It means that each year, the car retains 88% of its value from the previous year. Essentially, the car loses 12% of its value every year.
### Part b:
- [tex]\( f(9) \)[/tex] represents the value of the car 9 years after it was purchased. By substituting [tex]\( t = 9 \)[/tex] into the function, we calculate the car's value at that time:
[tex]\[
f(9) = 24,500 \cdot (0.88)^9 \approx 7753.72
\][/tex]
So, after 9 years, the car's value is approximately [tex]$7,753.72.
### Part c:
- To sketch a graph of the function \( f(t) = 24,500 \cdot (0.88)^t \), we focus on some specific points such as \( f(0) \), \( f(1) \), and \( f(2) \).
- \( f(0) \):
\[
f(0) = 24,500 \cdot (0.88)^0 = 24,500
\]
This is the initial value of the car.
- \( f(1) \):
\[
f(1) = 24,500 \cdot (0.88)^1 = 21,560
\]
After 1 year, the car's value is $[/tex]21,560.
- [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 24,500 \cdot (0.88)^2 = 18,972.8
\][/tex]
After 2 years, the car's value is approximately $18,972.80.
- Sketching the graph:
- The graph starts at [tex]\( (0, 24,500) \)[/tex], which represents the car's initial value.
- It then decreases to [tex]\( (1, 21,560) \)[/tex] and [tex]\( (2, 18,972.8) \)[/tex].
- The graph is an exponential decay curve, sloping downwards as [tex]\( t \)[/tex] increases, demonstrating the car depreciating over time.
By plotting these points and the smooth curve through them, you can visually represent the decrease in the car's value over the years.
### Part a:
- The number 24,500 represents the initial value of the car. This is the purchase price of the car in dollars at the beginning, or when [tex]\( t = 0 \)[/tex].
- The number 0.88 is the depreciation factor. It means that each year, the car retains 88% of its value from the previous year. Essentially, the car loses 12% of its value every year.
### Part b:
- [tex]\( f(9) \)[/tex] represents the value of the car 9 years after it was purchased. By substituting [tex]\( t = 9 \)[/tex] into the function, we calculate the car's value at that time:
[tex]\[
f(9) = 24,500 \cdot (0.88)^9 \approx 7753.72
\][/tex]
So, after 9 years, the car's value is approximately [tex]$7,753.72.
### Part c:
- To sketch a graph of the function \( f(t) = 24,500 \cdot (0.88)^t \), we focus on some specific points such as \( f(0) \), \( f(1) \), and \( f(2) \).
- \( f(0) \):
\[
f(0) = 24,500 \cdot (0.88)^0 = 24,500
\]
This is the initial value of the car.
- \( f(1) \):
\[
f(1) = 24,500 \cdot (0.88)^1 = 21,560
\]
After 1 year, the car's value is $[/tex]21,560.
- [tex]\( f(2) \)[/tex]:
[tex]\[
f(2) = 24,500 \cdot (0.88)^2 = 18,972.8
\][/tex]
After 2 years, the car's value is approximately $18,972.80.
- Sketching the graph:
- The graph starts at [tex]\( (0, 24,500) \)[/tex], which represents the car's initial value.
- It then decreases to [tex]\( (1, 21,560) \)[/tex] and [tex]\( (2, 18,972.8) \)[/tex].
- The graph is an exponential decay curve, sloping downwards as [tex]\( t \)[/tex] increases, demonstrating the car depreciating over time.
By plotting these points and the smooth curve through them, you can visually represent the decrease in the car's value over the years.
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