# Exponential Growth/Decay Calculator

An exponential growth/decay calculator is a tool that helps you calculate the growth or decay of a quantity over time using an exponential function. This calculator allows you to determine the final value or the rate of change based on the initial value, growth/decay rate, and time.

## What is Exponential growth/decay

Exponential growth and decay refer to the dynamic processes in which a quantity increases or decreases over time at an exponential rate. These concepts are commonly encountered in various fields, including biology, finance, population studies, and physics.

Exponential Growth: Exponential growth occurs when a quantity increases over time at a rate proportional to its current value. In other words, the growth rate of the quantity is directly influenced by its current size. As time progresses, the rate of growth accelerates due to the increasing base value. The growth pattern follows an exponential function.

Mathematically, exponential growth can be expressed using the formula:

A(t) = A₀ * e^(rt)

Where:

- A(t) represents the quantity at time t.
- A₀ is the initial quantity (often at t=0 or the starting point).
- e is Euler's number, approximately equal to 2.71828.
- r is the growth rate per unit of time.

Exponential Decay: Exponential decay refers to the process in which a quantity decreases over time at an exponential rate. Similar to exponential growth, the decay rate is proportional to the current value of the quantity. As time progresses, the rate of decay slows down due to the decreasing base value. The decay pattern also follows an exponential function.

Mathematically, exponential decay can be expressed using the formula:

A(t) = A₀ * e^(-rt)

Where:

- A(t) represents the quantity at time t.
- A₀ is the initial quantity or value.
- e is Euler's number.
- r is the decay rate per unit of time.

Both exponential growth and decay have practical applications in various scenarios. For example, exponential growth can model population growth, compound interest, or the spread of infectious diseases. Exponential decay models radioactive decay, the diminishing effect of drugs in the body, or the depreciation of assets over time.

Understanding exponential growth and decay allows for predicting future values, analyzing rates of change, and making informed decisions in relevant fields.

## Exponential growth/decay formula

*x*(*t*) = *x*_{0} × (1 + *r*)^{ t}

*x*(t) is the value at time t.

*x*_{0} 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/decay Example

*x*_{0} = 50

*r* = 4% = 0.04

*t* = 90 hours

*x*(*t*) = *x*_{0}* * × (1 + *r*)^{ t} = 50×(1+0.04)^{90} = 1706