Question

A car valued at $32,566 depreciates by 19% per year. What is the rate, r, to be used in the exponential model to estimate the value of the car after t years? (Note: r is the percentage in the decay factor, b=1−r. For decay, r is a negative value.) Enter your answer as a decimal rounded to the nearest hundredth.

Answer 1

To estimate the value of the car after t years using exponential decay, we need to find the decay factor, b. The decay factor is given by b = 1 - r, where r is the percentage in the decay rate. In this case, the car depreciates by 19% per year, so the decay rate is -19%. The rate, r, to be used in the exponential model is approximately -0.209.

Step-by-step explanation:

To estimate the value of the car after t years using exponential decay, we need to find the decay factor, b. The decay factor is given by b = 1 - r, where r is the percentage in the decay rate. In this case, the car depreciates by 19% per year, so the decay rate is -19%.

To find the decay factor, we can use the formula r = ln(1 + p), where p is the growth rate in decimal form. Since the car is depreciating, we use a negative growth rate: p = -0.19. Plugging in the values, we get:

r = ln(1 + (-0.19))

Solving for r using a calculator, we find r ≈ -0.209

Therefore, the rate, r, to be used in the exponential model to estimate the value of the car after t years is approximately -0.209.