Exponential Decay Calculator

How to Calculate Exponential Decay

Exponential decay is a mathematical concept that describes how quantities decrease over time at a rate proportional to the current amount. This process is common in many natural and scientific phenomena, such as radioactive decay, population decline, and the cooling of objects.

Formula

The general formula for exponential decay is:

\[ A = A_0 (1 - r)^t \]

Where:

• A is the final value after decay
• A₀ is the initial value
• r is the decay rate (as a decimal)
• t is the time elapsed

Calculation Steps

1. Identify the initial value (A₀), decay rate (r), and time (t).
2. Convert the decay rate to a decimal if it's given as a percentage.
3. Subtract the decay rate from 1: (1 - r).
4. Raise (1 - r) to the power of t.
5. Multiply the result by the initial value (A₀).
6. The result is the final value after decay.

Example

Let's calculate the exponential decay of a substance with an initial amount of 100 grams, decaying at a rate of 20% per hour, after 3 hours:

1. Initial value (A₀) = 100 grams, Decay rate (r) = 20% = 0.20, Time (t) = 3 hours
2. A = 100 (1 - 0.20)³
3. A = 100 (0.80)³
4. A = 100 × 0.512
5. A = 51.2 grams

After 3 hours, 51.2 grams of the substance remain.

Visual Representation

This diagram illustrates the exponential decay of the substance over 3 hours, showing how the amount decreases more rapidly at first and then slows down as the quantity becomes smaller.