To determine the amount of a radioactive sample remaining after a given time, you can use the concept of half-life and the radioactive decay equation. The decay equation for a radioactive substance is:
N(t) = N₀ * (1/2)^(t / T₁/₂)
where: N(t) is the amount remaining after time t, N₀ is the initial amount, t is the elapsed time, and T₁/₂ is the half-life of the substance.
In this case, the half-life of polonium-210 (Po-210) is given as 138 days. We want to find the amount remaining after 30 days, starting with a 50 mg sample. Let's calculate it:
N₀ = 50 mg (initial amount) t = 30 days (elapsed time) T₁/₂ = 138 days (half-life)
N(t) = 50 mg * (1/2)^(30 / 138)
To calculate this, divide the elapsed time (30) by the half-life (138), then raise 1/2 to that power:
N(t) = 50 mg * (1/2)^(0.2174)
Using a calculator, you can evaluate this:
N(t) ≈ 50 mg * 0.800
N(t) ≈ 40 mg
Therefore, approximately 40 mg of the original 50 mg sample of polonium-210 will remain after 30 days.