Question

the value of a used car, \\(c(t)\\), is modeled by the function \\(c(t) = 3,200(1 - 0.42)^t\\), where \\(t\\) is time in years. does this function represent exponential growth or decay? what is the percent rate of change?

it represents exponential growth, and the percent rate of change is 42%.
it represents exponential growth, and the percent rate of change is 58%.
it represents exponential decay, and the percent rate of change is 42%.
it represents exponential decay, and the percent rate of change is 58%.

Explanation:

Identify the model type

$$ c(t) = 3200(1 - 0.42)^t $$

Since the base is \(1 - 0.42 = 0.58\), which is less than \(1\), the function represents exponential decay.

Determine the percent rate of change

The decay rate \(r\) is given by:

$$ r = 0.42 $$

Converting this to a percentage:

$$ 0.42 \times 100\% = 42\% $$

Match with the correct option

The function represents exponential decay, and the percent rate of change is \(42\%\).

Answer:

• It represents exponential growth, and the percent rate of change is 42%.
• It represents exponential growth, and the percent rate of change is 58%.
• It represents exponential decay, and the percent rate of change is 42%. (Correct answer)
• It represents exponential decay, and the percent rate of change is 58%.