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