To determine whether the decreasing value of the car can be modeled by a linear or an exponential decay function, we need to analyze how the value changes each year.
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Linear Decay: In a linear decay function, the value changes by a constant absolute amount each year. This would imply the depreciation is the same dollar amount every year.
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Exponential Decay: In an exponential decay function, the value decreases by a constant percentage each year. This means that each year's value is a percentage of the previous year's value, leading to a multiplicative rate of change.
Let’s examine the values:
- Year 0: $54,000
- Year 1: $51,840 (decreases by $2,160)
- Year 2: $49,766 (decreases by $2,074)
- Year 3: $47,776 (decreases by $1,990)
- Year 4: $45,865 (decreases by $1,911)
The amount of depreciation is not constant. The first year drops $2,160, but the amounts decrease in subsequent years.
Next, let’s calculate the percentage decrease from year to year:
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Year 0 to Year 1: \[ \left(\frac{54000 - 51840}{54000}\right) \times 100 \approx 4% \]
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Year 1 to Year 2: \[ \left(\frac{51840 - 49766}{51840}\right) \times 100 \approx 4% \]
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Year 2 to Year 3: \[ \left(\frac{49766 - 47776}{49766}\right) \times 100 \approx 4% \]
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Year 3 to Year 4: \[ \left(\frac{47776 - 45865}{47776}\right) \times 100 \approx 4% \]
Since the car's value decreases by approximately the same percentage each year (around 4%), this demonstrates exponential decay.
The correct answer is:
Exponential decay function; the value decreases by 4% per year, which is a constant percent rate of change.
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