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

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 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

exponential decay function; the value decreases by $6,000 per year, which is a constant percent rate of change

1 answer

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.

  1. 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.

  2. 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.