Question

The half-life of a radioactive kind of barium is 3 minutes. If you start with 9,152 grams of it, how much will be left after 12 minutes?

Answer 1

To determine how much of the radioactive barium will be left after 12 minutes, we can use the half-life formula. The half-life of the substance is the time it takes for half of it to decay.

Given:
- Initial amount (N₀) = 9,152 grams
- Half-life (t₁/₂) = 3 minutes
- Time elapsed (t) = 12 minutes

First, we need to find out how many half-lives have passed in 12 minutes. We can calculate this by dividing the total time by the half-life of the substance:

\[
\text{Number of half-lives} = \frac{t}{t_{1/2}} = \frac{12 \text{ minutes}}{3 \text{ minutes}} = 4
\]

Next, we use the formula for exponential decay, which states that the remaining quantity after \( n \) half-lives is given by:

\[
N = N_0 \left(\frac{1}{2}\right)^n
\]

Now, substituting in the values we have:

\[
N = 9,152 \left(\frac{1}{2}\right)^4
\]

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

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

Now substituting this back into our equation:

\[
N = 9,152 \times \frac{1}{16}
\]

Now, calculating:

\[
N = 9,152 \div 16 = 572
\]

Thus, after 12 minutes, the amount of the radioactive barium left will be **572 grams**.