Exponential Growth/Decay Calculator

Exponential growth and decay can be modeled using the formula \( x(t) = x_0 \times (1 + r)^t \), where \( x(t) \) represents the value after time \( t \), \( x_0 \) is the initial value, \( r \) is the growth (or decay) rate per time period, and \( t \) is the number of time periods.

Calculation Formulas

1. Exponential Growth: If \( r \) is positive, the formula represents exponential growth.
2. Exponential Decay: If \( r \) is negative, the formula represents exponential decay.

Example Calculation

Exponential Growth

Suppose you invest $1000 (\( x_0 = 1000 \)) in a savings account with an annual interest rate of 5% (\( r = 0.05 \)). To find the balance after 3 years (\( t = 3 \)), use the formula: \[ x(3) = 1000 \times (1 + 0.05)^3 \]

Exponential Decay

As an example of decay, consider a radioactive substance with an initial amount of 100 grams (\( x_0 = 100 \)) that decays at a rate of 2% per year (\( r = -0.02 \)). To find the remaining amount after 5 years (\( t = 5 \)), use the formula: \[ x(5) = 100 \times (1 - 0.02)^5 \]

Importance and Use Cases

1. Finance: Calculating compound interest, investment growth, or loan amortization.
2. Population Dynamics: Modeling population growth or decline.
3. Physics and Chemistry: Understanding radioactive decay or chemical reaction rates.
4. Biology: Modeling population growth of species or spread of diseases.

FAQ

1. What's the difference between \( e^{rt} \) and \( (1 + r)^t \) in exponential calculations? \( e^{rt} \) is used for continuous growth or decay, while \( (1 + r)^t \) is for discrete or periodic growth or decay.

2. Can \( r \) be greater than 1? Yes, but it represents a growth rate of over 100% per period, which is uncommon in most real-world scenarios.

3. How do you determine the time period? The time period \( t \) depends on the context – it could be years, months, days, etc., and should match the rate \( r \).

4. What if \( t \) is not a whole number? \( t \) can be a fraction or decimal, especially in cases like calculating growth for part of a year.

5. Is the formula different for money compared to populations or chemicals? The core formula remains the same, but interpretation and units may differ based on the context.

Exponential growth and decay models are powerful tools for understanding and predicting change over time in a wide range of real-world situations.