EEE - Exponential Decay Models (Lesson)

Exponential Decay Models

You pay $12,000 for a car. The value then depreciates at a rate of 15% per year. How much will your car be worth in 3 years?

Let's start out by making a table:

Years Owned

Value

0

$12,000

1

12,000 - 0.15(12,000) = 12,000 - 1,800 = $10,200

2

10,200 - 0.15(10,200) = 10,200 - 1,530 = $8,670

3

8,670 - 0.15(8,670) = 8,670 - 1,300.50 = $7,369.50

Just like with Jane's trading cards, we are taking a percentage of the previous value - but this time we are subtracting it! That is because the value is depreciating or decreasing by a constant rate! Let's determine what that rate is:

10,200/12,000 = 0.85

8,670/10,200 = 0.85

7,369.50/8,670 = 0.85

So, the previous year's value is being multiplied by 0.85 each year, or the car is keeping 85% of its value each year. So each year, we are taking 1 whole and subtracting 15% or 0.15.

We call this an exponential decay model and the general formula is:

image explaining exponential decay  

Let's check out the equation and graph of the growth of Jane's playing card:

Graph 
function image 

Answer these questions about the model:

  1. What is the y-intercept of the graph? What does it represent?
  2. Predict the value of the car after 10 years?
  3. In how many years will the value of the car be about $5,000?

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

Watch this video to explore a few more exponential decay relationships:

 

  

Exponential Decay Models Practice

  1. Mr. Dale buys a car for $18,500. The value depreciates 9% per year. Write an exponential decay function for this situation. 
  2. How much will Mr. Dale's car be worth after 3 years? 
  3. Approximate when the car will be worth about $10,000?
  4. The half-life of a particular element is 10 days. If you begin with 40 grams of the element write an exponential decay function that models the decay of the element. Let t =1 represent one half life (or 10 days). 
  5. How much of the element is left about 40 days?

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

IMAGES CREATED BY GAVS