1. What is the Exponential Growth Formula?

The exponential growth formula describes how quantities increase over time when the growth rate is proportional to the current amount. It's widely used in population growth, finance, physics, and biology.

2. How Does the Calculator Work?

The calculator uses the exponential growth equation:

Where:

• \( A \) — Final amount (unitless)
• \( A_0 \) — Initial amount (unitless)
• \( k \) — Growth rate (1/time)
• \( t \) — Time (time units)
• \( e \) — Euler's number (~2.71828, unitless)

Explanation: The formula shows how an initial quantity grows exponentially at a constant rate over time.

3. Applications of Exponential Growth

Details: This model applies to population growth, compound interest, radioactive decay (with negative k), bacterial growth, and many natural phenomena.

4. Using the Calculator

Tips: Enter initial amount (must be ≥0), growth rate (can be positive or negative), and time (must be ≥0). The calculator will compute the final amount.

5. Frequently Asked Questions (FAQ)

Q1: What does a negative growth rate mean?
A: A negative k value indicates exponential decay rather than growth, commonly seen in radioactive decay processes.

Q2: How is this different from linear growth?
A: Exponential growth increases by percentages (faster over time), while linear growth adds fixed amounts each period.

Q3: What are typical units for the growth rate?
A: Common units include 1/year, 1/month, or 1/second, depending on the time units used.

Q4: Can I use this for financial calculations?
A: Yes, this models continuous compounding. For periodic compounding, use \( A = A_0(1 + r/n)^{nt} \).

Q5: What's the doubling time formula?
A: Doubling time = \( \ln(2)/k \) when k > 0. For 10% growth rate (k=0.1), doubling time ≈ 6.93 time units.