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 how the value changes over the years.
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Check for Linear Decay: In a linear decay function, the value would decrease by a constant amount each year.
- Year 0 to Year 1: $54,000 - $51,840 = $2,160
- Year 1 to Year 2: $51,840 - $49,766 = $2,074
- Year 2 to Year 3: $49,766 - $47,776 = $1,990
- Year 3 to Year 4: $47,776 - $45,865 = $1,911
The amount by which the value decreases is not constant; it decreases less each year. Therefore, it is not a linear decay function.
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Check for Exponential Decay: In an exponential decay function, the value decreases by a constant percentage each year.
- From Year 0 to Year 1: $\frac{51,840}{54,000} \approx 0.96 \Rightarrow \text{decrease of 4%}$
- From Year 1 to Year 2: $\frac{49,766}{51,840} \approx 0.96 \Rightarrow \text{decrease of 4%}$
- From Year 2 to Year 3: $\frac{47,776}{49,766} \approx 0.96 \Rightarrow \text{decrease of 4%}$
- From Year 3 to Year 4: $\frac{45,865}{47,776} \approx 0.96 \Rightarrow \text{decrease of 4%}$
The value consistently decreases by about 4% each year, indicating a constant percent rate of change.
Given this analysis, the correct response is:
exponential decay function; the value decreases by 4% per year, which is a constant percent rate of change.
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