# Half-Life

How to find the half-life of a substance: formula, 1 examples, and its solution.

## Formula

### Formula

The half-life is the amount of time
when a value continuously decreases
to one half.
A0 → A0/2

To find the half-life,
use the continuous exponential decay formula.
Set A0 = A0 and A = A0/2.

A0ert = A0/2

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

The simpler formula is
-rt = ln 2.
(This is used in science and engineering.)

But, in high school math,
it's good to set A = A0/2
and use A0ert = A0/2 formula.

## Example

### Solution

It says
find the half-life of the substance.

So set
A0 = A0 and A = A0/2.

The weight decreases
at a rate of 4% per second.

So r = -0.04/second.

The minus sign means decreasing.

A0 = A0
A = A0/2
r = -0.04

The weight decreases continuously.

Then A0⋅e-0.04⋅t = A0/2.

The goal is to find the time t.

Divide both sides by A0.

e-0.04t = 1/2

Then -0.04t = ln 1/2.

Logarithmic Form

Natural Logarithm

ln 2-1 = -1 ln 2

Logarithm of a Power

Multiply -1 to both sides.

It says
assume ln 2 = 0.69.

Then 0.04t = 0.69.

Divide both sides by 0.04.

Move the decimal points
2 digits to the right.

0.69/0.04 = 69/4

The unit of the time is [second].

So write 69/4 seconds.

So 69/4 seconds is the answer.