Continuous Exponential Decay: Time

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

Formula

Formula

Just like finding the time of
continuous exponential growth,

to find the time of
continuous exponential decay,

use the exponential change formula.

A0ert = A

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

Example

Example

Solution

The initial value of the weight is 50g.

So A0 = 50g.

It says
after how many seconds will the weight
be less than 20g?

So set A = 20g.

The weight decreases
at a rate of 3% per second.

So r = -0.03/second.

The minus sign means decreasing.

A0 = 50
A = 20
r = -0.03

The weight decreases continuously.

Then 50⋅e-0.03⋅t = 20.

The goal is to find the time t.

Divide both sides by 50.

20/50
= 2/5
= 4/10
= 0.4

e-0.03t = 0.4

Then -0.03t = ln 0.4.

Logarithmic Form

Natural Logarithm

It says
assume ln 0.4 = -0.916.

Then -0.03t = -0.916.

Multiply -1 to both sides.

Divide both sides by 0.03.

Move the decimal points
2 digits to the right.

0.916/0.03 = 91.6/3

Find the value of 91.6/3
to the ones.

91.6/3 = 30.xx

t = 30.xx
Round this up to the nearest ones:
30.xx → 31.

The unit of the time is [second].

So write
After 31 seconds.

t = 30.xx means
after 30.xx seconds,
the weight will be exactly 20g.

So after 31 seconds,
the weight will be less than 20g.

So
after 31 seconds
is the answer.