Score: \( 4 / 15 \) Penalty: 1 off Question An element with mass 210 grams decays by \( 8.3 \% \) per minute. How much of the element is remaining after 15 minutes, to the nearest 10 oth of a gram? Answer Attempt 1 out of 2

To find the amount of the element remaining after 15 minutes, we can use the formula for exponential decay: \[ A(t) = A_0 \times (1 - r)^t \] where: - \( A(t) \) is the amount remaining after time \( t \), - \( A_0 \) is the initial amount, - \( r \) is the decay rate per minute, - \( t \) is the time in minutes. Given: - Initial mass \( A_0 = 210 \) grams, - Decay rate \( r = 8.3\% = 0.083 \), - Time \( t = 15 \) minutes. Substitute the values into the formula: \[ A(15) = 210 \times (1 - 0.083)^{15} \] Now, we can calculate the amount remaining after 15 minutes. Calculate the value by following steps: - step0: Calculate: \(210\left(1-0.083\right)^{15}\) - step1: Subtract the numbers: \(210\times 0.917^{15}\) - step2: Convert the expressions: \(210\left(\frac{917}{1000}\right)^{15}\) - step3: Simplify: \(210\times \frac{917^{15}}{1000^{15}}\) - step4: Rewrite the expression: \(10\times 21\times \frac{917^{15}}{1000^{15}}\) - step5: Rewrite the expression: \(10\times 21\times \frac{917^{15}}{10^{45}}\) - step6: Reduce the numbers: \(21\times \frac{917^{15}}{10^{44}}\) - step7: Multiply: \(\frac{21\times 917^{15}}{10^{44}}\) The amount of the element remaining after 15 minutes is approximately 57.24787 grams. Rounded to the nearest 10th of a gram, the amount remaining is 57.2 grams.