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 exponental function. \[ A(t)=647\left(\frac{1}{2}\right)^{\frac{t}{30}} \] Find the initial amount in the sample and the amount remaining after 80 years. Round your answers to the nearest gram as necessary.

The initial amount in the cesium-137 sample is simply the value of \( A(0) \). Plugging \( t = 0 \) into the equation: \[ A(0) = 647\left(\frac{1}{2}\right)^{\frac{0}{30}} = 647 \times 1 = 647 \, \text{grams} \] So, the initial amount is 647 grams. Now, to find the amount remaining after 80 years, plug \( t = 80 \) into the equation: \[ A(80) = 647\left(\frac{1}{2}\right)^{\frac{80}{30}} \] Calculate \( \frac{80}{30} \) which is approximately 2.67, so we raise \( \frac{1}{2} \) to that power: \[ A(80) = 647 \times \left(\frac{1}{2}\right)^{2.67} \] Which is roughly \( A(80) = 647 \times 0.157 \) (using a calculator for \( \left(\frac{1}{2}\right)^{2.67} \)). This gives: \[ A(80) \approx 102.8 \, \text{grams} \] Rounding to the nearest gram, we find that the amount remaining after 80 years is approximately 103 grams. To summarize: The initial amount is 647 grams, and the amount remaining after 80 years is approximately 103 grams.