Exponential Growth Calculator

About Exponential Growth Calculator

Exponential growth: quantity increases by a constant percentage per time period. Doubling time = ln(2)/r.

How to use this calculator

1. Enter the starting value, usually written as \(N_0\).
2. Enter the growth or decay rate \(r\) as a decimal, so 5% becomes 0.05 and 8% decay becomes -0.08.
3. Enter the time \(t\) in the units used by the rate, such as years or months.
4. Check the result \(N(t)\), which gives the amount after time \(t\).

The formula explained

The formula \(N(t) = N_0 \cdot e^{rt}\) computes the amount after time \(t\) when a quantity changes continuously at rate \(r\). If \(r\) is positive, the amount grows, and if \(r\) is negative, it decays.

• N(t) = the amount after time \(t\)
• \(N_0\) = the starting amount at time 0
• r = the continuous growth rate or decay rate, written as a decimal
• t = time elapsed

Step by step method

1. Start with the initial amount \(N_0\).
2. Multiply by \(e^{rt}\), where \(r\) is the rate and \(t\) is the time.
3. Evaluate the exponent and multiply to find the final amount.

Worked example

Problem. A bacteria culture starts with 500 cells and grows continuously at 3% per hour. How many cells will there be after 4 hours?

1. Use the formula: \(N(t) = 500\cdot e^{0.03\cdot 4}\).
2. Compute the exponent: \(0.03\cdot 4 = 0.12\), so \(N(t) = 500\cdot e^{0.12}\).
3. Find the value: \(e^{0.12} \approx 1.1275\), so \(N(t) \approx 500\cdot 1.1275 = 563.75\).

Tips and common mistakes

• Make sure the rate is in decimal form, not percent form, so use 0.03 instead of 3%.
• Use a negative value for decay, such as \(-0.08\) for 8% continuous decrease.

Frequently asked questions