Ountrinic Epronamial und Lepricimic forctions Evelhating an exponential function that models a feul worid sitimition 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 tyears is given by the following exponential function. \[ A(t)=621\left(\frac{1}{2}\right)^{\frac{t}{30}} \] Find the initial amount in the sample and the amount remaining after 100 years. Round your answers to the nearest gram as necessary. Initial amount: grams Amount after 100 years: \( \square \) grams

The initial amount of cesium-137 in the sample can be found by evaluating the function at \( t = 0 \). Plugging this into the equation gives: \[ A(0) = 621\left(\frac{1}{2}\right)^{\frac{0}{30}} = 621\left(1\right) = 621 \text{ grams} \] To find the amount remaining after 100 years, substitute \( t = 100 \) into the function: \[ A(100) = 621\left(\frac{1}{2}\right)^{\frac{100}{30}} \approx 621\left(0.0285\right) \approx 17.67 \text{ grams} \] After rounding, the amount remaining after 100 years is approximately 18 grams. So, the answers are: Initial amount: 621 grams Amount after 100 years: 18 grams