Question
Use the table to answer the question.
The table lists the value of a car each year after it is purchased.
Year 0 1 2 3 4
Value ($) 54,000 51,840 49,766 47,776 45,865
Determine whether the decreasing value can be modeled by a linear decay function or an exponential decay function, and why.
(1 point)
Responses
exponential decay function; the value decreases by 4% per year, which is a constant percent rate of change
exponential decay function; the value decreases by 4% per year, which is a constant percent rate of change
linear decay function; the value decreases by 0.96 per year, which is a constant percent rate of change
linear decay function; the value decreases by 0.96 per year, which is a constant percent rate of change
linear decay function; the value decreases by 96% per year, which is a constant percent rate of change
linear decay function; the value decreases by 96% per year, which is a constant percent rate of change
exponential decay function; the value decreases by $6,000 per year, which is a constant percent rate of change
exponential decay function; the value decreases by $6,000 per year, which is a constant percent rate of change
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The table lists the value of a car each year after it is purchased.
Year 0 1 2 3 4
Value ($) 54,000 51,840 49,766 47,776 45,865
Determine whether the decreasing value can be modeled by a linear decay function or an exponential decay function, and why.
(1 point)
Responses
exponential decay function; the value decreases by 4% per year, which is a constant percent rate of change
exponential decay function; the value decreases by 4% per year, which is a constant percent rate of change
linear decay function; the value decreases by 0.96 per year, which is a constant percent rate of change
linear decay function; the value decreases by 0.96 per year, which is a constant percent rate of change
linear decay function; the value decreases by 96% per year, which is a constant percent rate of change
linear decay function; the value decreases by 96% per year, which is a constant percent rate of change
exponential decay function; the value decreases by $6,000 per year, which is a constant percent rate of change
exponential decay function; the value decreases by $6,000 per year, which is a constant percent rate of change
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Answers
GPT-4o mini
To determine whether the decreasing value of the car can be modeled by a linear decay function or an exponential decay function, let's analyze the data provided:
- The value of the car decreases from **$54,000** in Year 0 to **$51,840** in Year 1.
- The value decreases to **$49,766** in Year 2.
- It continues decreasing to **$47,776** in Year 3 and **$45,865** in Year 4.
Calculating the percentage change each year:
1. From Year 0 to Year 1:
\[
\frac{51,840 - 54,000}{54,000} = \frac{-4,160}{54,000} \approx -0.077
\]
which is approximately **-7.7%**.
2. From Year 1 to Year 2:
\[
\frac{49,766 - 51,840}{51,840} = \frac{-2,074}{51,840} \approx -0.040
\]
which is approximately **-4.0%**.
3. From Year 2 to Year 3:
\[
\frac{47,776 - 49,766}{49,766} = \frac{-1,990}{49,766} \approx -0.040
\]
which is approximately **-4.0%**.
4. From Year 3 to Year 4:
\[
\frac{45,865 - 47,776}{47,776} = \frac{-1,911}{47,776} \approx -0.040
\]
which is approximately **-4.0%**.
The percentage decreases are not constant, but they are relative to the value of the car each year. This indicates that the decay is not linear (where a constant amount is subtracted each year), but rather follows an exponential decay pattern, where a constant percentage is deducted based on the value at the beginning of each year.
Thus, the correct response is:
**exponential decay function; the value decreases by 4% per year, which is a constant percent rate of change.**
- The value of the car decreases from **$54,000** in Year 0 to **$51,840** in Year 1.
- The value decreases to **$49,766** in Year 2.
- It continues decreasing to **$47,776** in Year 3 and **$45,865** in Year 4.
Calculating the percentage change each year:
1. From Year 0 to Year 1:
\[
\frac{51,840 - 54,000}{54,000} = \frac{-4,160}{54,000} \approx -0.077
\]
which is approximately **-7.7%**.
2. From Year 1 to Year 2:
\[
\frac{49,766 - 51,840}{51,840} = \frac{-2,074}{51,840} \approx -0.040
\]
which is approximately **-4.0%**.
3. From Year 2 to Year 3:
\[
\frac{47,776 - 49,766}{49,766} = \frac{-1,990}{49,766} \approx -0.040
\]
which is approximately **-4.0%**.
4. From Year 3 to Year 4:
\[
\frac{45,865 - 47,776}{47,776} = \frac{-1,911}{47,776} \approx -0.040
\]
which is approximately **-4.0%**.
The percentage decreases are not constant, but they are relative to the value of the car each year. This indicates that the decay is not linear (where a constant amount is subtracted each year), but rather follows an exponential decay pattern, where a constant percentage is deducted based on the value at the beginning of each year.
Thus, the correct response is:
**exponential decay function; the value decreases by 4% per year, which is a constant percent rate of change.**