Exponential Growth/Decay Calculator

Online exponential growth/decay calculator.

Exponential growth/decay formula

x(t) = x₀ × (1 + r) ^t

x(t) is the value at time t.

x₀ is the initial value at time t=0.

r is the growth rate when r>0 or decay rate when r<0, in percent.

t is the time in discrete intervals and selected time units.

Exponential growth calculator

Enter the initial value x₀, growth rate r and time interval t and press the = button:

Enter initial value (x0):
Enter growth/decay rate (r):		%
Enter time (t):
Value at time t (x(t)):

Example

x₀ = 50

r = 4% = 0.04

t = 90 hours

x(t) = x₀ × (1 + r) ^t = 50×(1+0.04)⁹⁰ = 1706

Exponential Growth and Decay

The Exponential Growth/ Decay Calculator is used to solve exponential growth/decay problems. It will calculate any one of the values from the other three in the exponential decay model equation. One of the most prevalent applications of exponential functions involves growth and decay models. Exponential growth and decay show up in a host of natural applications. From population growth and continuously compounded interest to radioactive decay and Newton’s law of cooling, exponential functions are ubiquitous in nature.

Exponential growth/decay formula

x(t) = x₀ × (1 + r)^t

x(t) is the value at time t.

x0 is the initial value at time t=0.

r is the growth rate when r > 0 or decay rate when r < 0, in percent.

t is the time in discrete intervals and selected time units.

How to calculate exponential growth

Consider the following problem: the population of a small city at the beginning of 2019 was 10,000 people. It was noticed that the population of the city grows at a steady rate of 5% annually. What should you do to calculate the projected population size in the year 2030? From the given data, we can conclude the initial population value, x₀, equals 10,000. Also, we have the growth rate of r = 5%.

Therefore, the exponential growth formula we should use is:

x(t) = 10,000 * (1 + 0.05)^t = 10,000 * 1.05^t.

Here t is the number of years passed since 2019. In our case, for the year 2030, we should use t = 11, since this is the difference in the number of years between 2030 and the initial year 2019. Finally, we get:

x(11) = 10,000 * 1.05¹¹ = 17,103.

So, the projected number of inhabitants of our small city in the year 2030 is around 17,103.

Example on how to use the formula for exponential decay

Radioactive decay is a well-known example of where the exponential decay formula is used. For a given initial quantity of radioactive substance, you may write down the law which governs its decay over time. But, maybe a more fun example is to measure how much coffee remains in your body at 10 pm if you drank a cup of coffee with x⁰ = 95 mg of caffeine at noon.

We will use the fact that the half-life of caffeine in the human body is approximately six hours. Half-life is defined as the time needed a given quantity to reduce to half of its initial value. So, in this example we have

x(6) = 1/2 * 95 mg = 47.5 mg.

Here, it will be easier to use the alternative notation for the exponential growth formula:

x(t) = 95 * ek*t.

 

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