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Math

Exponential Decay Calculator.

Calculates the remaining quantity of a substance after a given time, based on its initial amount, decay rate, and elapsed time.

Results update live as you type.
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Your inputs

How it works

  1. 1

    Enter the initial amount of the substance.

  2. 2

    Enter the decay rate as a percentage per time unit.

  3. 3

    Enter the elapsed time in the same time units.

  4. 4

    The calculator computes the remaining amount using the exponential decay formula.

initial_amount * exp(-decay_rate/100 * time_elapsed)

Frequently asked questions

What is exponential decay?

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value, following the formula A = A0 * e^(-kt).

How do I interpret the decay rate?

The decay rate is the percentage decrease per time unit. For example, a 5% decay rate means the quantity decreases by 5% of its current value each time unit.

Can I use this for half-life calculations?

Yes, you can find the half-life by setting the remaining amount to half the initial amount and solving for time, but this calculator directly computes the remaining amount for a given time.

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Results

Formula checked

Remaining Amount

60.653units

Decayed Amount39.35units
Percentage Remaining60.65%
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Estimate for general guidance only — verify important decisions with an appropriate professional.

How it works

Calculates the remaining quantity of a substance after a given time, based on its initial amount, decay rate, and elapsed time.

  1. Enter the initial amount of the substance.
  2. Enter the decay rate as a percentage per time unit.
  3. Enter the elapsed time in the same time units.
  4. The calculator computes the remaining amount using the exponential decay formula.

Formulas

The math behind this calculator, written out so you can verify the result.

Exponential Decay Formula

A(t) = A0 * e^(-kt)

A(t) is the amount after time t, A0 is the initial amount, k is the decay constant (rate/100), and e is Euler's number.

Example:

Input: A0 = 100, k = 0.05, t = 10

Calculation: 100 * e^(-0.05*10) = 100 * e^(-0.5)

Result: ≈ 60.65

Real-world use cases

Where this calculation shows up in everyday life.

Radioactive Decay

Estimate the remaining mass of a radioactive isotope after a certain period.

Example: Carbon-14 dating

Drug Clearance

Model how a drug concentration decreases in the bloodstream over time.

Example: Determine dosage intervals

Depreciation

Calculate the value of an asset that depreciates at a constant percentage rate.

Example: Car value after 5 years

Tips and common mistakes

Tips

  • Ensure the time units match the decay rate units (e.g., if rate is per year, time must be in years).
  • For half-life, use the formula t_half = ln(2) / k, where k = rate/100.
  • Use a small decay rate for long-term processes to avoid extremely small results.
  • Check that the initial amount is positive; negative values are not physically meaningful.

Common Mistakes to Avoid

  • Forgetting to convert the percentage decay rate to a decimal (divide by 100).
  • Using the decay rate as a decimal directly without dividing by 100.
  • Confusing decay rate with half-life; they are related but not the same.

Assumptions and limitations

  • Use the stated inputs and units.
  • Results are estimates for planning and education.
  • Check measurements and source data before making an important decision.