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Exponential functions don't only represent significant growth, but also consistent decline over specific periods of time. This consistent decline is called exponential decay.
Exponential decay is the process of value reduction at a consistent rate. Mathematically, an exponential decay function can be written as:
With the conditions:
When , the value of becomes smaller as the value of increases, causing the function graph to decrease.
The graph of an exponential decay function shows a decreasing rate of decline. Initially, the function value decreases rapidly, then the rate of decrease slows down as it approaches zero but never actually reaches zero.
Unlike exponential growth graphs that rise at an increasing rate, exponential decay graphs fall at a decreasing rate.
The graph below shows an exponential decay function with base and coefficient , which can be expressed as:
Exponential decay has many applications in real life. Here are some examples:
A patient receives a dose of pain medication. Every hour, half of the remaining drug dose will decay and be eliminated from the body.
If we model the amount of drug remaining after as , then:
Calculating the remaining dose:
With this pattern, we can calculate that after , the amount of the drug in the patient's body will be less than .
Radioactive substances decay at a certain rate, typically expressed as a percentage per unit of time. If we have an initial mass and a decay rate per unit time, then the mass remaining after time can be expressed as:
When a ball is dropped, each time it bounces, the bounce height reaches a portion of its previous height. If the initial height is and each bounce reaches a proportion of the previous height, then the height of the -th bounce is:
For example, if a basketball is dropped from a height of and each bounce reaches of the previous height, then:
The bounce height function can be expressed as:
The ball will eventually stop bouncing when the bounce height becomes very small, around at the th bounce.
Here are some practice problems to deepen your understanding of exponential decay:
Two hundred mg of a substance is injected into the body of a patient suffering from lung cancer. The substance is excreted from the body through the kidneys every hour. If of the substance is eliminated from the patient's body every hour, how many mg of the substance remains in the patient's body after ?
The mass of a radioactive substance is at am. The decay rate of the radioactive substance is per hour. How much radioactive substance will remain later?
A basketball is dropped from a height of . The ball hits the ground and then bounces back to a height of of its previous height. The ball continues to bounce with the same height ratio until it finally stops bouncing and falls to the ground.
Let be the amount of substance remaining after . We have:
After :
So, the substance remaining after is .
Let be the mass of the radioactive substance after . We have:
For the basketball:
The bounce height function is:
In mathematical modeling, exponential decay is used to model various phenomena such as:
Beyond scientific applications, exponential concepts can also be found in social contexts. For example, the relationship between charity (giving) and blessing (receiving) can be illustrated with an exponential growth function. The more we share with others in need, the more goodness will return, at an increasingly faster rate.
After :
The ball will stop bouncing when its height approaches zero. Based on calculations, at the th bounce, the ball's height reaches , which is small enough to consider the ball has stopped bouncing. However, considering the ball's mass, it might have stopped bouncing by the th bounce, when its height is about .
| Bounce | Height |
|---|---|
| | | | | | | | |
