Questions: In 2002, a car was purchased for 19,000. Each year since, the resale value has decreased by 28%. Let t be the number of years since 2002. Let y be the value of the car, in dollars. Write an exponential function showing the relationship between y and t.

Solution Steps

We are given that a car was purchased for \$19,000 in 2002, and its resale value decreases by 28% each year. We need to express the relationship between the car's value \( y \) and the number of years \( t \) since 2002 using an exponential function.

The general formula for exponential decay is:

$$y = a \cdot (1 - r)^t$$

where:

• $a$ is the initial value,
• $r$ is the rate of decay,
• $t$ is the time in years.

In this problem:

• The initial value $a$ is \$19,000.
• The rate of decay $r$ is 28%, or 0.28 in decimal form.

Substituting these values into the formula gives:

$$y = 19000 \cdot (1 - 0.28)^t$$

Simplifying the expression inside the parentheses:

$$y = 19000 \cdot (0.72)^t$$

Final Answer