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?

To determine how much radioactive barium remains after 12 minutes, we'll use the concept of **half-life**, which is the time it takes for half of a radioactive substance to decay. **Given:** - **Half-life (\( t_{1/2} \))** = 3 minutes - **Initial amount (\( A_0 \))** = 9,152 grams - **Total time (\( t \))** = 12 minutes **Steps:** 1. **Calculate the number of half-lives that have passed:** \[ \text{Number of half-lives} = \frac{t}{t_{1/2}} = \frac{12 \text{ minutes}}{3 \text{ minutes}} = 4 \] 2. **Use the half-life formula to find the remaining amount:** \[ A = A_0 \times \left(\frac{1}{2}\right)^n \] where \( n \) is the number of half-lives. \[ A = 9152 \times \left(\frac{1}{2}\right)^4 = 9152 \times \frac{1}{16} = \frac{9152}{16} = 572 \text{ grams} \] **Answer:** After 12 minutes, **572 grams** of the radioactive barium will remain.