- Difficulty
- Standard
- Suggested time
- 5 min
- Marks
- 3
Exponential growth and decay practice question 4
A radioactive isotope decays so that 30 % of any sample is gone after 10 days. How long does it take for a 200 g sample to decay to 50 g?
Give the answer in days.
Two unknowns here: the decay constant and the time. How much of any sample is left after 10 days, and which of the two does that fact pin down?
Write . The 30 % fact says , which determines .
Then solve , i.e. . Both steps are logarithms.
An exact expression, or a decimal to two decimal places.
Theory UBC Math 100 §11.3CLP-1 §3.3Key results
Worked solution
Strategy
Two measurements, two logarithms: the first fixes the decay constant from the 30 % fact, the second finds the time for the sample to reach a quarter of its mass.
Working
Losing 30 % leaves 70 %. Since 0.7 < 1, log 0.7 is negative, so k < 0 as decay requires.
log(1/4) = −log 4 and log 0.7 = −log(10/7); the signs cancel.
The exact form is the answer; the decimal is only a sense of scale.
Answer
The answer is 10 log 4 over log of 10/7 days, about 38.9 days.
Check
After 40 days the sample is 200 · 0.7⁴ ≈ 48 g, just under 50 g, so reaching 50 g takes a little under 40 days.
Where this goes wrong
Reasoning linearly: “30 % per 10 days, so 75 % takes 25 days”. Decay is proportional to what remains, so each 10 days removes 30 % of a smaller amount and the process slows down.
Leans on
Solving an exponential equation with logarithms, from Functions, graphs and inverses
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