Question

How much of a radioactive kind of bismuth will be left after 8 minutes if the half-life is 2 minutes and you start with 79,360 grams?
grams

Answer 1

To solve the problem of how much radioactive bismuth will remain after 8 minutes, given a half-life of 2 minutes and an initial quantity of 79,360 grams, we can use the half-life formula. The formula for remaining quantity \( N \) after time \( t \) is given by:

\[
N = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}}
\]

where:
- \( N_0 \) is the initial quantity (79,360 grams in this case),
- \( t \) is the total time (8 minutes),
- \( T_{1/2} \) is the half-life (2 minutes).

First, we need to determine how many half-lives fit into the 8 minutes:

\[
\text{Number of half-lives} = \frac{t}{T_{1/2}} = \frac{8 \text{ minutes}}{2 \text{ minutes}} = 4
\]

Next, we substitute \( N_0 \), the number of half-lives, and calculate:

\[
N = 79,360 \times \left( \frac{1}{2} \right)^{4}
\]

Calculating \( \left( \frac{1}{2} \right)^{4} \):

\[
\left( \frac{1}{2} \right)^{4} = \frac{1}{16}
\]

Now we can find the remaining quantity:

\[
N = 79,360 \times \frac{1}{16}
\]

Calculating \( 79,360 \div 16 \):

\[
79,360 \div 16 = 4,960
\]

Thus, the amount of radioactive bismuth left after 8 minutes is:

\[
\boxed{4960} \text{ grams}
\]