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Exponential Decay Calculator

Calculate exponential decay with interactive visualizations, half-life computation, decay constant, and step-by-step solutions. Solve for initial amount, final amount, decay rate, or time.

Exponential Decay Calculator
P(t)
Final Amount
Calculate remaining
P₀
Initial Amount
Find original
r
Decay Rate
Find rate
t
Time
Calculate duration
Quick Examples: Radioactive Decay Find Initial Find Rate Half-Life
P₀ Initial Amount
Starting quantity
P(t) Final Amount
Amount after decay
r Decay Rate
Use decimal (0.05 = 5%)
t Time
Time periods elapsed
Precision
Result accuracy

Embed Exponential Decay Calculator Widget
Solve For: This field is required. Precision: This field is required.

About Exponential Decay Calculator

Welcome to the Exponential Decay Calculator, a comprehensive tool for solving exponential decay problems with step-by-step solutions and interactive visualizations. Whether you need to calculate the final amount after decay, determine the initial amount, find the decay rate, or compute the time required for decay, this calculator provides accurate results with detailed explanations.

What is Exponential Decay?

Exponential decay describes the decrease of a quantity at a rate proportional to its current value. Unlike linear decay where a fixed amount is lost each period, exponential decay removes a fixed percentage, causing the quantity to decrease more slowly as it gets smaller. This behavior is described by the continuous decay formula:

$$P(t) = P_0 \cdot e^{-rt}$$

Where:

Exponential Decay Calculator Features

Four Calculation Modes

This calculator can solve for any variable in the exponential decay equation:

Additional Calculations

Beyond the main result, the calculator also provides:

Interactive Decay Curve

The calculator generates a visual representation of the decay process, showing how the quantity decreases over time with the calculated point marked on the curve.

Time-Series Table

A detailed table shows the decay progression at regular time intervals, including amount remaining, amount decayed, and percentage remaining at each point.

How to Use This Calculator

  1. Select what to solve for: Choose what variable you want to calculate (Final Amount, Initial Amount, Decay Rate, or Time). The calculator will display the required input fields.
  2. Enter known values: Input the values you know. For decay rate, use decimal format (0.05 for 5%). All values must be positive.
  3. Select precision: Choose the number of decimal places for your result (4 to 10 decimal places).
  4. Calculate: Click the Calculate button to see your result with step-by-step solution, decay curve, and time-series table.
  5. Analyze results: Review the decay visualization and additional metrics like half-life and decay constant.

Understanding Half-Life

Half-life is the time required for a quantity to reduce to half its initial value. It is calculated using:

$$t_{1/2} = \frac{\ln(2)}{r} \approx \frac{0.693}{r}$$
Key insight: Half-life is constant regardless of the initial amount. After one half-life, 50% remains. After two half-lives, 25% remains. After three half-lives, 12.5% remains, and so on.

Decay Rate vs. Decay Constant

In the continuous decay formula P(t) = P₀e^(-rt), the decay rate r and decay constant λ (lambda) are equivalent. They represent how quickly the quantity decreases:

Real-World Applications

Radioactive Decay

Radioactive isotopes decay at rates characterized by their half-lives. Carbon-14 has a half-life of about 5,730 years, making it useful for dating organic materials up to about 50,000 years old.

Example: If you have 100g of a radioactive isotope with a half-life of 10 years, after 30 years (3 half-lives), you would have 100 × (1/2)³ = 12.5g remaining.

Drug Metabolism (Pharmacokinetics)

Drugs are eliminated from the body through exponential decay. The elimination half-life determines how often a drug needs to be administered to maintain therapeutic levels.

Asset Depreciation

Some financial models use exponential decay to model how assets like vehicles and electronics lose value over time.

Population Decline

Declining populations often follow exponential decay patterns when the death rate exceeds the birth rate by a consistent proportion.

Cooling and Heating (Newton's Law)

The temperature difference between an object and its environment decreases exponentially over time according to Newton's Law of Cooling.

Electrical Circuits

Capacitors discharge through resistors following exponential decay, characterized by the RC time constant.

Related Formulas

Solving for Different Variables

The exponential decay formula can be rearranged to solve for any variable:

$$P_0 = P(t) \cdot e^{rt}$$ $$r = -\frac{\ln(P(t)/P_0)}{t}$$ $$t = -\frac{\ln(P(t)/P_0)}{r}$$

Discrete vs. Continuous Decay

While this calculator uses continuous exponential decay (base e), discrete decay uses a different formula:

$$P(t) = P_0 \cdot (1 - r)^t$$

For continuous decay: P(t) = P₀e^(-rt)
For discrete decay: P(t) = P₀(1-r)^t

Frequently Asked Questions

What is exponential decay?

Exponential decay describes the decrease of a quantity at a rate proportional to its current value. It follows the formula P(t) = P₀ × e^(-rt), where P₀ is the initial amount, r is the decay rate, and t is time. Common examples include radioactive decay, drug metabolism, and depreciation.

How do I calculate final amount after decay?

To calculate the final amount, use the formula P(t) = P₀ × e^(-rt). Enter your initial amount P₀, decay rate r, and time t. The calculator multiplies the initial amount by e raised to the power of negative r times t to give you the remaining amount.

What is half-life in exponential decay?

Half-life is the time required for a quantity to reduce to half its initial value. It is calculated as t½ = ln(2) / r, where r is the decay rate. Half-life is constant regardless of the initial amount and is commonly used in radioactive decay and pharmacology.

What is the difference between decay rate and decay constant?

In continuous exponential decay using P(t) = P₀ × e^(-rt), the decay rate r and decay constant λ (lambda) are the same value. They represent how quickly the quantity decreases. A higher value means faster decay. The decay constant is often denoted by the Greek letter lambda.

What are real-world applications of exponential decay?

Exponential decay models many natural and financial phenomena including: radioactive decay of isotopes, drug concentration in the bloodstream over time, depreciation of assets, population decline, cooling of objects (Newton's law of cooling), discharge of capacitors in electronics, and decay of sound intensity.

How do I calculate decay rate from initial and final amounts?

Use the formula r = -ln(P(t)/P₀) / t. Divide the final amount by the initial amount, take the natural logarithm, divide by time, and negate the result. This gives you the decay rate per time unit.

What happens if my decay rate is negative?

A negative decay rate actually represents exponential growth, not decay. For true decay, the rate must be positive, meaning the quantity decreases over time. Use a positive decay rate for this calculator.

Additional Resources

For further learning about exponential decay:

Reference this content, page, or tool as:

"Exponential Decay Calculator" at https://MiniWebtool.com/exponential-decay-calculator/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: Jan 12, 2026

You can also try our AI Math Solver GPT to solve your math problems through natural language question and answer.

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