Question

the half - life of a radioactive isotope is the time it takes for a quantity of the isotope to be reduced to half its initial mass. starting with 120 grams of a radioactive isotope, how much will be left after 4 half - lives? use the calculator provided and round your answer to the nearest gram.

Explanation:

Step1: Recall half - life formula

The amount of a radioactive substance remaining after \(n\) half - lives is given by \(A = A_0\times(\frac{1}{2})^n\), where \(A_0\) is the initial amount and \(n\) is the number of half - lives. Here, \(A_0 = 120\) grams and \(n = 4\).

Step2: Substitute values

Substitute \(A_0 = 120\) and \(n = 4\) into the formula: \(A=120\times(\frac{1}{2})^4\).

Step3: Calculate \((\frac{1}{2})^4\)

\((\frac{1}{2})^4=\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}=\frac{1}{16}\).

Step4: Calculate the remaining amount

\(A = 120\times\frac{1}{16}=\frac{120}{16}=7.5\) grams. Rounding to the nearest gram, \(A\approx8\) grams.

Answer:

8 grams