10
advanced-math

A radioactive isotope decays so that 40% remains after 10 years. What is the approximate half-life?

A

About 7.5 years

B

5 years

C

10 years

D

4 years

Correct Answer: A

Choice A is the correct answer. Find half-life from remaining percentage.

  1. Model: A(t)=A0(0.5)t/hA(t) = A_0(0.5)^{t/h} where hh is half-life.
  2. Given: After 10 years, 40% remains: 0.40=(0.5)10/h0.40 = (0.5)^{10/h}.
  3. Solve: Take log: log(0.40)=10hlog(0.5)\log(0.40) = \frac{10}{h} \log(0.5).
  4. Calculate: h=10log(0.5)log(0.40)10(0.301)0.3987.56h = \frac{10 \log(0.5)}{\log(0.40)} \approx \frac{10(-0.301)}{-0.398} \approx 7.56 years.

💡 Strategic Tip: Use logarithms or test values: if h=7.5h=7.5, then (0.5)10/7.50.39(0.5)^{10/7.5} \approx 0.39 ✓.

Choice B is incorrect because after 10 years (2 half-lives), only 25% would remain. Choice C is incorrect because this would leave 50%, not 40%. Choice D is incorrect because this would leave about 18%.