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 ($) 54000 51840 49766 47776 45865
Determine whether the decreasing value can be modeled by a linear decay function or an exponential decay function, and why.
(1 point)
linear decay function; the value decreases by 0.96 per year, which is a constant percent rate of change
exponential decay function, the value decreases by $6,000 per year, which is a constant percent rate of change
exponential decay function; the value decreases by 4% per year, which is a constant percent rate of change
linear decay function; the value decreases by 96% per year, which is a constant percent rate of change
The table lists the value of a car each year after it is purchased
Year 0 1 2 3 4
Value ($) 54000 51840 49766 47776 45865
Determine whether the decreasing value can be modeled by a linear decay function or an exponential decay function, and why.
(1 point)
linear decay function; the value decreases by 0.96 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 4% 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
Answers
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Answered by
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, we need to analyze the changes in value over the years.
Looking at the values from the table:
- From Year 0 to Year 1: 54,000 to 51,840 (decrease of 3,160)
- From Year 1 to Year 2: 51,840 to 49,766 (decrease of 2,074)
- From Year 2 to Year 3: 49,766 to 47,776 (decrease of 1,990)
- From Year 3 to Year 4: 47,776 to 45,865 (decrease of 1,911)
The decreases are not constant; they are getting smaller each year, which indicates that the rate of change is not linear.
Now, looking at the percentage change:
- From Year 0 to Year 1: \( \frac{51,840 - 54,000}{54,000} \approx -0.05885 \) or about 5.88%
- From Year 1 to Year 2: \( \frac{49,766 - 51,840}{51,840} \approx -0.04016 \) or about 4.02%
- From Year 2 to Year 3: \( \frac{47,776 - 49,766}{49,766} \approx -0.03993 \) or about 3.99%
- From Year 3 to Year 4: \( \frac{45,865 - 47,776}{47,776} \approx -0.03993 \) or about 4.01%
The decreases are about 4% on average after Year 1, indicating a constant fractional rate of change rather than a fixed dollar amount.
Therefore, the correct conclusion is:
**Exponential decay function; the value decreases by about 4% per year, which is a constant percent rate of change.**
Looking at the values from the table:
- From Year 0 to Year 1: 54,000 to 51,840 (decrease of 3,160)
- From Year 1 to Year 2: 51,840 to 49,766 (decrease of 2,074)
- From Year 2 to Year 3: 49,766 to 47,776 (decrease of 1,990)
- From Year 3 to Year 4: 47,776 to 45,865 (decrease of 1,911)
The decreases are not constant; they are getting smaller each year, which indicates that the rate of change is not linear.
Now, looking at the percentage change:
- From Year 0 to Year 1: \( \frac{51,840 - 54,000}{54,000} \approx -0.05885 \) or about 5.88%
- From Year 1 to Year 2: \( \frac{49,766 - 51,840}{51,840} \approx -0.04016 \) or about 4.02%
- From Year 2 to Year 3: \( \frac{47,776 - 49,766}{49,766} \approx -0.03993 \) or about 3.99%
- From Year 3 to Year 4: \( \frac{45,865 - 47,776}{47,776} \approx -0.03993 \) or about 4.01%
The decreases are about 4% on average after Year 1, indicating a constant fractional rate of change rather than a fixed dollar amount.
Therefore, the correct conclusion is:
**Exponential decay function; the value decreases by about 4% per year, which is a constant percent rate of change.**
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