The radioactive substance uranium-240 has a half-life of 14 hours. The amount \( A(t) \) of a sample of uranium- 240 remaining (in grams) after \( t \) hours is given by the following exponential function. \( A(t)=5600\left(\frac{1}{2}\right)^{\frac{t}{14}} \) Find the initial amount in the sample and the amount remaining after 30 hours. Round your answers to the nearest gram as necessary. Initial amount: Amount after 30 hours: grams

The initial amount in the sample, when \( t = 0 \), can be found by substituting \( t = 0 \) into the function \( A(t) \): \[ A(0) = 5600\left(\frac{1}{2}\right)^{\frac{0}{14}} = 5600 \times 1 = 5600 \text{ grams.} \] Next, to find the amount remaining after 30 hours, substitute \( t = 30 \) into the function: \[ A(30) = 5600\left(\frac{1}{2}\right)^{\frac{30}{14}} = 5600\left(\frac{1}{2}\right)^{2.14} \approx 5600 \times 0.225 \approx 1266 \text{ grams.} \] So, the initial amount is 5600 grams and the amount remaining after 30 hours is approximately 1266 grams.