Understanding Exponential Decay

What is Exponential Decay?

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

Formula

The formula for exponential decay is:

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

Where:

• \( A \) is the final amount
• \( P \) is the initial principal amount
• \( r \) is the decay rate (in decimal form)
• \( t \) is the time period

Calculation Steps

Let's calculate the exponential decay for a radioactive substance:

1. Given:
   • Initial amount (\( P \)) = 100 grams
   • Decay rate (\( r \)) = 10% = 0.10
   • Time (\( t \)) = 5 years
2. Apply the exponential decay formula: \[ A = P(1 - r)^t \]
3. Substitute the known values: \[ A = 100(1 - 0.10)^5 \]
4. Perform the calculation: \[ A = 100 \times 0.59049 \] \[ A = 59.049 \]

Example and Visual Representation

Let's visualize exponential decay over time:

This graph illustrates:

• The initial value (100 grams) at the start
• The rapid decrease over time
• The final value (59.049 grams) after 5 years
• The characteristic curve of exponential decay