Final answer:
The half-life of a radioactive sample that decays down to almost nothing in 15 minutes can be calculated using the exponential decay formula. As the question seems to misstate the nature of radioactive decay, we use typical scenarios to estimate the number of half-lives within the given time frame and then derive the half-life time.
Step-by-step explanation:
To calculate the half-life of a radioactive sample that decays 78 times its original quantity in 15 minutes, we use the concept of half-life, which is the time taken for half of the radioactive nuclei in a sample to undergo decay. This decay process follows an exponential function, and the number of half-lives that can occur within a certain period can be determined by the formula:
N = N0(1/2)^(t/t1/2), where N is the final amount, N0 is the initial amount, t is the time elapsed, and t1/2 is the half-life of the substance.
To find the number of half-lives that fit into 15 minutes, we need to solve for n in the equation 1 = (1/2)^n where 1 represents the fraction of the original quantity remaining after n half-lives (since the sample decays 78 times its original quantity, practically nothing of the original quantity remains). Since this is not a typical scenario in which a sizable fraction remains, we interpret the '78 times' as a misunderstanding of the decay process, assuming instead that the student meant to convey multiple half-lives that approximate the total decay. Through logarithmic calculations, the number of half-lives in 15 minutes is found, and the half-life of the radioactive sample can be determined as 15 minutes divided by the number of half-lives.