1. **State the problem:** We have a radioactive substance with a half-life of 3 minutes. Starting with 9,152 grams, we want to find how much remains after 12 minutes. 2. **Formula used:** The amount remaining after time $t$ is given by the exponential decay formula: $$ A = A_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T}} $$ where: - $A_0$ is the initial amount, - $T$ is the half-life, - $t$ is the elapsed time, - $A$ is the amount remaining after time $t$. 3. **Plug in the values:** $$ A_0 = 9152, \quad T = 3, \quad t = 12 $$ 4. **Calculate the exponent:** $$ \frac{t}{T} = \frac{12}{3} = 4 $$ 5. **Calculate the remaining amount:** $$ A = 9152 \times \left(\frac{1}{2}\right)^4 $$ 6. **Simplify the power:** $$ \left(\frac{1}{2}\right)^4 = \frac{1}{2^4} = \frac{1}{16} $$ 7. **Multiply:** $$ A = 9152 \times \frac{1}{16} $$ 8. **Simplify the fraction:** $$ A = \cancel{9152} \times \frac{1}{\cancel{16}} = 572 $$ 9. **Final answer:** After 12 minutes, there will be **572 grams** of barium left.