# Continuous Exponential Decay: Final Value

How to find the final value of continuous exponential decay: formula, 1 example, and its solution.

## Formula

If a value shows

a continuous exponential change (growth or decay),

use this formula.

A = A_{0}e^{rt}

A: Final value

A_{0}: Initial value

e: Constant e

r: Rate of change (per time period)

t: Number of time period

## Example

The initial value of the weight is 80g.

So A_{0} = 80g.

The weight decreases at a rate of 5% per minute.

So r = -0.05/minute.

Write the unit [per minute].

The final value is the weight 1 [hour] later.

But the unit of the rate is [per minute].

So write 1 hour in [minutes]:

t = 60 minutes.

A_{0} = 80

r = -0.05

t = 60

The weight decreases continuously.

Then the expected weight A is

A = 80⋅e^{-0.05⋅60}.

-0.05⋅60 = -3

It says

assume e^{-3} = 0.05.

So 80⋅e^{-3} = 80⋅0.05.

80⋅0.05 = 4.0

The initial value A_{0} is in g.

So the final value A is 4.0g.

So 4.0g is the answer.