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Half-life

The half-life is the time until a quantity has halved. After one half-life half remains, after two a quarter remains, and so on. The number of halvings is the time divided by the half-life. The remainder is the starting amount divided by 2 raised to that number. The model is used for radioactive substances and for other decrease that halves at fixed intervals. A fixed percentage that is not 50 is not a halving.

remaining=a⋅(12)t/T\text{remaining} = a \cdot \left(\frac{1}{2}\right)^{t/T}after time t

Symbols

aastarting amount
TThalf-life
tttime

Example

Half-life 5 years, start 80 g, time 15 years.

The number of halvings is 15/5=315/5 = 3, and the remainder is 80/23=1080/2^3 = 10 g.

Three halvings: 80, 40, 20, 10.
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Part of Mathematics 2P: Exponential growth.