Continuous Exponential Decay: Final Value

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

Formula

Formula

If a value shows
a continuous exponential change (growth or decay),
use this formula.

A = A0ert

A: Final value
A0: Initial value
e: Constant e
r: Rate of change (per time period)
t: Number of time period

Exponential Decay: Final Value

Example

Example

Solution

The initial value of the weight is 80g.
So A0 = 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.

A0 = 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 A0 is in g.

So the final value A is 4.0g.

So 4.0g is the answer.