Mathspace-IM1B-LL-Task 4 1 3 4 8 eas is 2. A sample contains 400 grams of iodine-131, which has a half-life of 8 days. 2a) Write an expression for \( A \) to represent the amount of iodine-131 remaining in the sample after \( t \) days. \[ A=400(0.5)^{\frac{1}{8}} \] \[ \ldots \] 2b Find how much of the isotope would be left after 8 weeks. Give your answer correct to two decimal places. \[ A= \] \( \square \) Fnter your next step here grams

To find the amount of iodine-131 remaining in the sample after \( t \) days, we can use the formula for radioactive decay: \[ A = A_0 \times (0.5)^{\frac{t}{T}} \] where: - \( A \) is the amount of iodine-131 remaining after \( t \) days, - \( A_0 \) is the initial amount of iodine-131 (400 grams in this case), - \( t \) is the time in days, - \( T \) is the half-life of iodine-131 (8 days in this case). Given that the half-life of iodine-131 is 8 days, we can substitute the values into the formula: \[ A = 400 \times (0.5)^{\frac{t}{8}} \] Now, we need to find how much of the isotope would be left after 8 weeks. Since there are 7 days in a week, 8 weeks is equal to 56 days. We can substitute \( t = 56 \) into the formula to find the amount of iodine-131 remaining after 8 weeks: \[ A = 400 \times (0.5)^{\frac{56}{8}} \] Let's calculate this value. Calculate the value by following steps: - step0: Calculate: \(400\times 0.5^{\frac{56}{8}}\) - step1: Divide the terms: \(400\times 0.5^{7}\) - step2: Convert the expressions: \(400\left(\frac{1}{2}\right)^{7}\) - step3: Evaluate the power: \(400\times \frac{1}{128}\) - step4: Multiply: \(\frac{25}{8}\) The amount of iodine-131 remaining after 8 weeks is approximately 3.125 grams.