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. \( A(t)=266\left(\frac{1}{2}\right)^{\frac{t}{30}} \) Find the amount of the sample remaining after 20 years and after 100 years. Round your answers to the nearest gram as necessary. Amount after 20 years: Amount after 100 years: Dgrams

To find the amount of cesium-137 remaining after 20 years and 100 years, we can substitute the values of \( t \) into the function \( A(t) = 266\left(\frac{1}{2}\right)^{\frac{t}{30}} \). For \( t = 20 \): \[ A(20) = 266\left(\frac{1}{2}\right)^{\frac{20}{30}} = 266 \left(\frac{1}{2}\right)^{\frac{2}{3}} \approx 266 \times 0.39685 \approx 105.63 \text{ grams} \approx 106 \text{ grams} \] For \( t = 100 \): \[ A(100) = 266\left(\frac{1}{2}\right)^{\frac{100}{30}} = 266 \left(\frac{1}{2}\right)^{\frac{10}{3}} \approx 266 \times 0.08853 \approx 23.54 \text{ grams} \approx 24 \text{ grams} \] So, the amounts remaining are: Amount after 20 years: 106 grams Amount after 100 years: 24 grams