Final answer:
To find the time for 7/8 of a sample of gallium-65 to decay, we can use the formula for radioactive decay. Using the half-life of gallium-65, we can determine the time it takes for 7/8 of the sample to decay as approximately 1.324 minutes.
Step-by-step explanation:
To determine the time it would take for 7/8 of a sample of gallium-65 to decay, we can use the formula for radioactive decay. The half-life of gallium-65 is given as 15.2 minutes. Since radioactive decay follows first-order kinetics, we can use the formula N(t) = N(0) * (1/2)^(t/t1/2), where N(t) is the amount of the radioactive isotope remaining after time t, N(0) is the initial amount, and t1/2 is the half-life.
So, for 7/8 of the sample to decay, we want to find the time when N(t)/N(0) = 7/8. Plugging this into the formula we get (7/8) = (1/2)^(t/15.2). To solve for t, we can take the logarithm of both sides and solve for t: t = t1/2 * log(7/8) / log(1/2).
Plugging in the values t1/2 = 15.2 and log(7/8) / log(1/2) ≈ 0.0872, we find t ≈ 15.2 * 0.0872 ≈ 1.324 minutes.