# 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.

A_{0} → A_{0}/2

To find the half-life,

use the continuous exponential decay formula.

Set A_{0} = A_{0} and A = A_{0}/2.

A_{0}e^{rt} = A_{0}/2

A_{0}: 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 = A_{0}/2

and use A_{0}e^{rt} = A_{0}/2 formula.

## Example

### Example

### Solution

It says

find the half-life of the substance.

So set

A_{0} = A_{0} and A = A_{0}/2.

The weight decreases

at a rate of 4% per second.

So r = -0.04/second.

The minus sign means decreasing.

A_{0} = A_{0}

A = A_{0}/2

r = -0.04

The weight decreases continuously.

Then A_{0}⋅e^{-0.04⋅t} = A_{0}/2.

The goal is to find the time t.

Divide both sides by A_{0}.

e^{-0.04t} = 1/2

Then -0.04t = ln 1/2.

Logarithmic Form

Natural Logarithm

1/2 = 2^{-1}

Negative Exponent

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.