Continuous Exponential Growth: Time

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

Formula

Formula

Recall that
the continuous exponential change formula is
A = A0ert.

When finding the time t,
switch both sides
and use the 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 10g.

So A0 = 10g.

It says
after how many minutes will the weight
be more than 30g?

So set A = 30g.

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

So r = 0.05/minute.

A0 = 10
A = 30
r = 0.05

The weight increases continuously.

Then 10⋅e0.05⋅t = 30.

The goal is to find the time t.

Divide both sides by 10.

e0.05t = 3

Then 0.05t = ln 3.

Logarithmic Form

Natural Logarithm

It says
assume ln 3 = 1.099.

Then 0.05t = 1.099.

Divide both sides by 0.05.

Move the decimal points
2 digits to the right.

1.099/0.05 = 109.9/5

Find the value of 109.9/5
to the ones.

109.9/5 = 21.xx

t = 21.xx
Round this up to the nearest ones:
21.xx → 22.

The unit of the time is [minute].

So write
After 22 minutes.

t = 21.xx means
after 21.xx minutes,
the weight will be exactly 30g.

So after 22 minutes,
the weight will be more than 30g.

So
after 22 minutes
is the answer.