Question

the half - life of a radioactive kind of protactinium is 27 days. how much will be left after 54 days, if you start with 488 grams of it? grams

Explanation:

Step1: Determine the number of half - lives

The half - life of protactinium is 27 days. The total time passed is 54 days. To find the number of half - lives \(n\), we use the formula \(n=\frac{\text{Total time}}{\text{Half - life}}\). So \(n = \frac{54}{27}=2\).

Step2: Use the half - life formula for radioactive decay

The formula for the amount of a radioactive substance left after \(n\) half - lives is \(A = A_0\times(\frac{1}{2})^n\), where \(A_0\) is the initial amount, \(n\) is the number of half - lives, and \(A\) is the amount left.
We know that \(A_0 = 488\) grams and \(n = 2\). Substitute these values into the formula:
\(A=488\times(\frac{1}{2})^2\)
First, calculate \((\frac{1}{2})^2=\frac{1}{4}\). Then, \(A = 488\times\frac{1}{4}\).
\(488\times\frac{1}{4}=\frac{488}{4}=122\).

Answer:

122