The half-life of a substance is the amount of time it takes for half of the substance to decay.
If an initial population of size P has a half-life of d years (or any other unit of time), then the formula to find the final number A in t years is given by
A = P(1/2)^{t/d}
Problem 1 :
The half-life of carbon-14 is approximately 6000 years. How much of 800 g of this substance will remain after 30,000 years?
Solution :
Half-Life Decay Formula :
A = P(1/2)^{t/d}
Substitute.
P = 800
t = 30000
d = 6000
Then,
A = 800(1/2)^{30000/6000}
= 800(1/2)^{5}
= 800(0.5)^{5}
= 800(0.03125)
= 25
So, 25 g of carbon-14 will remain after 30,000 years.
Problem 2 :
A certain radioactive substance has a half-life of 12 days. This means that every 12 days, half of the original amount of the substance decays. If there are 128 milligrams of the radioactive substance today, how many milligrams will be left after 48 days?
Solution :
Half-Life Decay Formula :
A = P(1/2)^{t/d}
Substitute.
P = 128
t = 48
d = 12
Then,
A = 128(1/2)^{48/12}
= 128(0.5)^{4}
= 128(0.0625)
= 8
So, 8 milligrams of radio active substance will be left after 48 days.
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