Find the value of the car after 5 years and after 13 years. Round your answers to the nearest dollar as necessary.
Understand the Problem
The question is asking us to find the dollar value of a car at two specific ages, 5 years and 13 years, using the given exponential decay function. We will calculate the values using the formula provided.
Answer
The car is valued at $12,948 after 5 years and $5,866 after 13 years.
Answer for screen readers
After 5 years, the value of the car is approximately $12,948. After 13 years, the value is approximately $5,866.
Steps to Solve
- Substitute t for 5
To find the value of the car after 5 years, plug in $t = 5$ into the function:
$$ v(5) = 24,500 \times (0.88)^5 $$
- Calculate v(5)
Now calculate the value:
$$ v(5) = 24,500 \times (0.88)^5 = 24,500 \times 0.5277 $$
This gives:
$$ v(5) \approx 12,947.65 $$
- Round to the nearest dollar
Round the value to the nearest dollar:
$$ v(5) \approx 12,948 $$
- Substitute t for 13
Next, find the value of the car after 13 years by substituting $t = 13$:
$$ v(13) = 24,500 \times (0.88)^{13} $$
- Calculate v(13)
Now calculate the value:
$$ v(13) = 24,500 \times (0.88)^{13} = 24,500 \times 0.2390 $$
This results in:
$$ v(13) \approx 5,866.15 $$
- Round to the nearest dollar
Round this value to the nearest dollar:
$$ v(13) \approx 5,866 $$
After 5 years, the value of the car is approximately $12,948. After 13 years, the value is approximately $5,866.
More Information
These calculations illustrate how exponential decay affects the value of a car over time. The exponential function $v(t) = 24,500(0.88)^t$ shows that the value decreases rapidly, particularly in the earlier years. The factor of 0.88 indicates a depreciation rate of 12% per year.
Tips
- Forgetting to round the final answer: Always remember to round the final result to the nearest dollar.
- Miscalculating the exponent: Ensure correct calculations when computing values like $(0.88)^5$ and $(0.88)^{13}$.
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