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 ($) 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

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 $6,000 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 0.96 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

linear decay function, the value decreases by 96% per year, which is a constant percent rate of change

Answer 1

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.

1. **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.

2. **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.**