# Nakafa Framework: LLM

URL: /en/subject/high-school/10/mathematics/exponential-logarithm/exponential-decay
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/exponential-logarithm/exponential-decay/en.mdx

Output docs content for large language models.

---

import { FunctionChart } from "@repo/design-system/components/contents/function-chart";

export const metadata = {
  title: "Exponential Decay",
  description: "Discover exponential decay through drug elimination, radioactive decay & bouncing balls. Master decreasing functions with real-world applications.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/04/2025",
  subject: "Exponential and Logarithm",
};

## Definition of Exponential Decay

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:

<BlockMath math="f(x) = n \times a^x" />

With the conditions:

- <InlineMath math="0 < a < 1" /> (base)
- <InlineMath math="n" /> is a non-zero real number (coefficient)
- <InlineMath math="x" /> is any real number (variable)

When <InlineMath math="0 < a < 1" />, the value of <InlineMath math="a^x" /> becomes smaller as the value of <InlineMath math="x" /> increases, causing the function graph to decrease.

## Characteristics of Exponential Decay Graph

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 <InlineMath math="a = 0.5" /> and coefficient <InlineMath math="n = 50" />, which can be expressed as:

<BlockMath math="f(x) = 50 \times \left(\frac{1}{2}\right)^x" />

<div className="mt-4">
  <FunctionChart
    a={0.5}
    p={50}
    title="Exponential Decay Graph"
    description="Graph decreases at a progressively slower rate."
  />
</div>

## Applications of Exponential Decay

Exponential decay has many applications in real life. Here are some examples:

### Drug Elimination in the Body

A patient receives a 50 microgram 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 <InlineMath math="x" /> hours as <InlineMath math="f(x)" />, then:

<BlockMath math="f(x) = 50 \times \left(\frac{1}{2}\right)^x" />

Calculating the remaining dose:

- Initial dose: <InlineMath math="f(0) = 50" /> micrograms
- After 1 hour: <InlineMath math="f(1) = 50 \times \frac{1}{2} = 25" /> micrograms
- After 2 hours: <InlineMath math="f(2) = 25 \times \frac{1}{2} = 12.5" /> micrograms
- After 3 hours: <InlineMath math="f(3) = 12.5 \times \frac{1}{2} = 6.25" /> micrograms

With this pattern, we can calculate that after 9 hours, the amount of the drug in the patient's body will be less than 0.1 micrograms.

### Radioactive Decay

Radioactive substances decay at a certain rate, typically expressed as a percentage per unit of time. If we have an initial mass <InlineMath math="M_0" /> and a decay rate <InlineMath math="r" /> per unit time, then the mass remaining after time <InlineMath math="t" /> can be expressed as:

<BlockMath math="M(t) = M_0 \times (1-r)^t" />

### Ball Bounce

When a ball is dropped, each time it bounces, the bounce height reaches a portion of its previous height. If the initial height is <InlineMath math="h_0" /> and each bounce reaches a proportion <InlineMath math="p" /> of the previous height, then the height of the <InlineMath math="n" />-th bounce is:

<BlockMath math="h_n = h_0 \times p^n" />

For example, if a basketball is dropped from a height of 3 meters and each bounce reaches <InlineMath math="\frac{3}{5}" /> of the previous height, then:

- Initial height: <InlineMath math="h_0 = 300" /> cm
- First bounce: <InlineMath math="h_1 = 300 \times \frac{3}{5} = 180" /> cm
- Second bounce: <InlineMath math="h_2 = 180 \times \frac{3}{5} = 108" /> cm
- Third bounce: <InlineMath math="h_3 = 108 \times \frac{3}{5} = 64.8" /> cm

The bounce height function can be expressed as:

<BlockMath math="h(n) = 300 \times \left(\frac{3}{5}\right)^n" />

The ball will eventually stop bouncing when the bounce height becomes very small, around 0.65 cm at the 12th bounce.

## Exercises

Here are some practice problems to deepen your understanding of exponential decay:

1. 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 50% 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 5 hours?

2. The mass of a radioactive substance is 0.3 kg at 10 am. The decay rate of the radioactive substance is 15% per hour. How much radioactive substance will remain 8 hours later?

3. A basketball is dropped from a height of 3 meters. The ball hits the ground and then bounces back to a height of <InlineMath math="\frac{3}{5}" /> of its previous height. The ball continues to bounce with the same height ratio until it finally stops bouncing and falls to the ground.
   a. Draw a graph of the function showing the change in bounce height until the ball finally touches the ground.
   b. At which bounce does the ball finally stop bouncing?

### Answer Key

1. Let <InlineMath math="f(x)" /> be the amount of substance remaining after <InlineMath math="x" /> hours. We have:

   <BlockMath math="f(x) = 200 \times \left(\frac{1}{2}\right)^x" />

   After 5 hours:

   <BlockMath math="f(5) = 200 \times \left(\frac{1}{2}\right)^5 = 200 \times 0.03125 = 6.25" />

   So, the substance remaining after 5 hours is 6.25 mg.

2. Let <InlineMath math="f(x)" /> be the mass of the radioactive substance after <InlineMath math="x" /> hours. We have:

   <BlockMath math="f(x) = 0.3 \times (0.85)^x" />

   After 8 hours:

   <BlockMath math="f(8) = 0.3 \times (0.85)^8 \approx 0.083 \text{ kg} \text{ or about } 83 \text{ grams}." />

3. a. The bounce height function is:

   <BlockMath math="h(n) = 300 \times \left(\frac{3}{5}\right)^n" />

   <div className="mt-4">
     <FunctionChart
       a={3 / 5}
       p={300}
       title="Exponential Decay Graph"
       description="Graph decreases at a progressively slower rate."
     />
   </div>

   b. The ball will stop bouncing when its height approaches zero. Based on calculations, at the 12th bounce, the ball's height reaches 0.653035 cm, which is small enough to consider the ball has stopped bouncing. However, considering the ball's mass, it might have stopped bouncing by the 10th bounce, when its height is about 1.8 cm.

   | Bounce                       | Height                         |
   | ---------------------------- | ------------------------------ |
   | <InlineMath math="n = 0" />  | <InlineMath math="300" />      |
   | <InlineMath math="n = 1" />  | <InlineMath math="180" />      |
   | <InlineMath math="n = 2" />  | <InlineMath math="108" />      |
   | <InlineMath math="n = 3" />  | <InlineMath math="64.8" />     |
   | <InlineMath math="n = 4" />  | <InlineMath math="38.88" />    |
   | <InlineMath math="n = 5" />  | <InlineMath math="23.328" />   |
   | <InlineMath math="n = 6" />  | <InlineMath math="13.9968" />  |
   | <InlineMath math="n = 7" />  | <InlineMath math="8.39808" />  |
   | <InlineMath math="n = 8" />  | <InlineMath math="5.038848" /> |
   | <InlineMath math="n = 9" />  | <InlineMath math="3.023309" /> |
   | <InlineMath math="n = 10" /> | <InlineMath math="1.813985" /> |
   | <InlineMath math="n = 11" /> | <InlineMath math="1.088391" /> |
   | <InlineMath math="n = 12" /> | <InlineMath math="0.653035" /> |
   | <InlineMath math="n = 13" /> | <InlineMath math="0.391821" /> |
   | <InlineMath math="n = 14" /> | <InlineMath math="0.235092" /> |
   | <InlineMath math="n = 15" /> | <InlineMath math="0.141055" /> |

## Application of Concepts in Life

In mathematical modeling, exponential decay is used to model various phenomena such as:

- Radioactive decay
- Cooling of hot objects
- Depreciation of goods
- Drug elimination in the body

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.
