# Continuous Exponential Growth: Time

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

## Formula

Recall that

the continuous exponential change formula is

A = A_{0}e^{rt}.

When finding the time t,

switch both sides

and use the formula.

A_{0}e^{rt} = A

A_{0}: Initial value

r: Rate of change (per time period)

t: Number of time period

A: Final value

## Example

The initial value of the weight is 10g.

So A_{0} = 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.

A_{0} = 10

A = 30

r = 0.05

The weight increases continuously.

Then 10⋅e^{0.05⋅t} = 30.

The goal is to find the time t.

Divide both sides by 10.

e^{0.05t} = 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.