The radioactive substance cesium-137 has a half-life of 30 years. The amount \( A(t) \) (in grams) of a sample of cesium- 137 remaining after \( t \) years is given by the following exponential function. \( A(t)=621\left(\frac{1}{2}\right)^{\frac{t}{30}} \) Find the amount of the sample remaining after 20 years and after 50 years. Round your answers to the nearest gram as necessary. Amount after 20 years: Amount after 50 years:

To find the amount of cesium-137 remaining after 20 and 50 years, we'll use the given exponential function \( A(t) = 621\left(\frac{1}{2}\right)^{\frac{t}{30}} \). **For 20 years:** \[ A(20) = 621\left(\frac{1}{2}\right)^{\frac{20}{30}} \] Calculating \( \left(\frac{1}{2}\right)^{\frac{2}{3}} \) gives approximately \( 0.39685 \). Now multiply: \[ A(20) \approx 621 \times 0.39685 \approx 246.39 \] Rounding to the nearest gram, we get **246 grams** remaining after 20 years. **For 50 years:** \[ A(50) = 621\left(\frac{1}{2}\right)^{\frac{50}{30}} \] Calculating \( \left(\frac{1}{2}\right)^{\frac{5}{3}} \) gives approximately \( 0.25000 \). Now multiply: \[ A(50) \approx 621 \times 0.25000 \approx 155.25 \] Rounding to the nearest gram, we get **155 grams** remaining after 50 years. In summary, after 20 years, there are **246 grams** of cesium-137, and after 50 years, there are **155 grams** remaining.