Final answer:
The calculation of decay time for a radioactive element is done by understanding its half-life, the period in which half of it decays. To find the duration until only a desired amount remains, the number of half-lives that pass must be determined using the initial and final amounts, and the half-life. The student's question lacked specific values to make the calculation.
Step-by-step explanation:
The student has asked how long it will take for a sample of element X with a given half-life to decrease to a certain amount. The calculation of the time it takes for a radioactive sample to decay to a specified mass involves understanding the concept of a half-life, which is the amount of time it takes for half of the radioactive isotope to decay.
Using the formula N = N0 (1/2)(t/T), where N is the remaining amount, N0 is the initial amount, t is the time elapsed, and T is the half-life of the substance, we can find the number of half-lives that have passed to reduce the sample to the desired amount. Doubling the number of half-lives n will leave us with (1/2)n of the original sample. For instance, if n is 3, we will have (1/2)3 or 1/8 of the original sample.
In the provided question, the exact values for the initial amount, final amount, and the half-life of element X are not specified, and they are necessary to perform the calculation. However, once these values are known, the process described above can be used to determine the time it will take for the initial sample to decay to the final amount. For example, if the half-life of element X is 100 years, then every 100 years, the amount of element X will be reduced by half.