Question

Exponential decay occurs when the quantity of something decreases at a rate that is proportional to its current value. For example, a population of animals might decay as their habitat is slowly eroded by human encroachment, or the amount of radiation that is emitted from a radioactive substance could decrease as time passes.

Suppose that your friend Rhyley has saved $5,000 toward the cost of a car and her parents have agreed to match that amount, dollar for dollar. You learned in your algebra 2 class that the value of a car decreases as time passes. You want to use your knowledge to help your friend buy the best car that she can with the money she has available. Rhyley has narrowed the options down to two different cars, both of which have similar features:



Car A

Car B

Cost: $8,750

Mileage: 89,325

Year of Manufacture: 2015

Decay Factor (average): 12%

Cost: $9,995

Mileage: 73,498

Year of Manufacture: 2017

Decay Factor (average): 18%



Use the information given and your knowledge of exponential decay to help your friend Rhyley by answering the questions below.



1. Write an exponential decay function in the form for each of Car A and Car B. Explain how you determined the value of b for each function.

Answer 1

To model the depreciation of a car's value over time, exponential decay functions for Car A and Car B are created using the formula P(t) = P_0 e^{(kt)}. Car A's function is P_A(t) = 8750 e^{(-0.12t)}and Car B's function is P_B(t) = 9995 e^{(-0.18t)}, reflecting their decay rates of 12% and 18% respectively.

Step-by-step explanation:

Exponential decay can be used to model the depreciation of a car's value over time. The general exponential decay function is given by P(t) = P_0 e^{(kt)}, where P(t) is the value of the car after time t, P_0 is the initial value of the car, and k is the decay constant.

For Car A with an initial cost of $8,750 and an average decay factor of 12%, the decay function can be written as P_A(t) = 8750 e^{(-0.12t)}. The value of b, which is the base of the natural logarithm e, reflects the continuous decay rate of 12% per time period. For Car B, with an initial cost of $9,995 and an average decay factor of 18%, the decay function can be written as P_B(t) = 9995 e^{(-0.18t)}, similarly reflecting its continuous decay rate of 18% per time period

Each car's value will exponentially decrease over time according to its specific decay rate. Rhyley will need to consider the depreciation rates when determining which car will offer the best value for money in the long term.