<meta http-equiv="refresh" content="1; url=/nojavascript/"> Exponential Decay ( Real World ) | Algebra | CK-12 Foundation

# Exponential Decay

%
Best Score
Practice Exponential Decay
Best Score
%

## Real World Applications – Algebra I

### Topic

What’s a car worth?

### Student Exploration

#### How much is a car worth?

You’ve already done several activities in this concept that calculate a car’s worth. You already know that as time passes, a car’s value depreciates. How can you calculate the depreciation rate if it’s not given to you?

1. Choose a car that you, a friend, a relative, or a neighbor has. Make sure you can talk to the car’s owner. You have a lot of questions to ask them!
b. What is the car’s mileage?
2. Once you have this information, use the internet to find out how much this car is worth present day.
a. You can use the Kelly Blue Book website: www.kbb.com
3. Create an equation using the information you have.
a. What is the initial value?
b. What is the amount of time that has passed?
c. What is the value of the car for that amount of time passed?
4. What are some methods to find out the depreciation rate?
a. One of the methods is using a calculator. You can use a the Guess and Check method to find out different values of “$b$” that would make the equation true.
b. How could you find the “$b$” value algebraically? Show all steps. You will need a scientific or graphing calculator to do this.
5. Graph this relationship. What values are on the $x-$axis and $y-$axis? What values make sense on these axes?
6. Is there ever a time when the car will have no value? If so, when will this happen? If not, why not?

### Extension Investigation

7. Use the internet to find the value of a different car in two different years.
8. Use this information to find the initial value of the car.
a. Need help? Get the unknown variable on one side of the equation for both equations, and set the two equations equal to each other.
b. You will need to use logarithms to solve this!

### Resources Cited

This is an example of a car and finding out the depreciation value over time. Of course, these estimates are going to be approximate because all cars’ values depend on a number of variables, as you might have seen on the Kelly Blue Book website. A 2004 Jetta Wagon that has 100,000 miles on it in 2012 is worth $6,200. When it was purchased new in 2004, it was worth$18,000. We can find out how much the car’s value depreciated over time, assuming that approximately the same number of miles were driven on the car every year.

We know that the initial value is $18,000. We also know that from 2004 to 2012, eight years have passed. After 8 years, the value of the car is at$6,200. We can use all of this information to write the equation for exponential decay and find the depreciation value.

$Y = ab^t$, or we can use $y = a(1-b)^t$, since we’re finding the depreciation rate, which is decreasing.

$6202 = 18000(1-b)^8$ First we divide both sides by 18,000, then take the eighth root of both sides. The “$b$” value is represented by 1 – the result, or 0.1247. To turn this into a percentage, the depreciation rate is 12.47% per year.