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 look at how the value changes year over year.
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Check for linear decay: In a linear decay function, the value decreases by a constant amount each year. Checking the value changes:
- 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 of decrease is not constant, so it is not linear decay.
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Check for exponential decay: In an exponential decay function, the value decreases by a constant percentage each year. To see if this is true, we calculate the percentage decrease for each year:
- Year 0 to Year 1: ($54,000 - $51,840) / $54,000 = $2,160 / $54,000 ≈ 0.04 (4%)
- Year 1 to Year 2: ($51,840 - $49,766) / $51,840 = $2,074 / $51,840 ≈ 0.04 (4%)
- Year 2 to Year 3: ($49,766 - $47,776) / $49,766 ≈ 0.04 (4%)
- Year 3 to Year 4: ($47,776 - $45,865) / $47,776 ≈ 0.04 (4%)
The percentage decrease is consistent at approximately 4% each year.
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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