Question

how much of a radioactive kind of actinium will be left after 132 years if you start with 301,184 grams and the half - life is 22 years?

grams

Explanation:

Step1: Find the number of half - lives

The formula to find the number of half - lives \(n\) is \(n=\frac{t}{T}\), where \(t\) is the total time and \(T\) is the half - life. Here, \(t = 132\) years and \(T=22\) years. So \(n=\frac{132}{22}=6\).

Step2: Use the radioactive decay formula

The radioactive decay formula is \(N = N_0\times(\frac{1}{2})^n\), where \(N_0\) is the initial amount, \(n\) is the number of half - lives, and \(N\) is the remaining amount. We know that \(N_0 = 301184\) grams and \(n = 6\). Then \(N=301184\times(\frac{1}{2})^6\). Since \((\frac{1}{2})^6=\frac{1}{64}\), we have \(N=\frac{301184}{64}=4706\) grams.

Answer:

4706