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. \[ A(t)=4700\left(\frac{1}{2}\right)^{\frac{t}{14}} \] Find the initial amount in the sample and the amount remaining after 60 hours. Round your answers to the nearest gram as necessary.

The initial amount of the sample is simply the value of \( A(t) \) when \( t = 0 \). Plugging in \( t = 0 \) into the function: \[ A(0) = 4700\left(\frac{1}{2}\right)^{\frac{0}{14}} = 4700\left(\frac{1}{2}\right)^0 = 4700 \] So, the initial amount in the sample is 4700 grams. Now, to find the amount remaining after 60 hours, we substitute \( t = 60 \) into the function: \[ A(60) = 4700\left(\frac{1}{2}\right)^{\frac{60}{14}} \] Calculating \( \frac{60}{14} \approx 4.2857 \): \[ A(60) = 4700\left(\frac{1}{2}\right)^{4.2857} \] Now, calculating \( \left(\frac{1}{2}\right)^{4.2857} \approx 0.0517 \): \[ A(60) \approx 4700 \times 0.0517 \approx 242.99 \] Rounding to the nearest gram, the amount remaining after 60 hours is approximately 243 grams.