An element with mass 970 grams decays by \( 27.7 \% \) per minute. How much of the element is remaining after 13 minutes, to the nearest 1oth of a gram?

To find the amount of the element remaining after 13 minutes, we can use the formula for exponential decay: \[ A = A_0 \times (1 - r)^t \] where: - \( A \) is the amount remaining after time \( t \), - \( A_0 \) is the initial amount, - \( r \) is the decay rate per minute, and - \( t \) is the time in minutes. Given: - Initial amount \( A_0 = 970 \) grams, - Decay rate \( r = 27.7\% = 0.277 \) per minute, - Time \( t = 13 \) minutes. Substitute these values into the formula and calculate the amount remaining after 13 minutes. Calculate the value by following steps: - step0: Calculate: \(970\left(1-0.277\right)^{13}\) - step1: Subtract the numbers: \(970\times 0.723^{13}\) - step2: Convert the expressions: \(970\left(\frac{723}{1000}\right)^{13}\) - step3: Simplify: \(970\times \frac{723^{13}}{1000^{13}}\) - step4: Rewrite the expression: \(10\times 97\times \frac{723^{13}}{1000^{13}}\) - step5: Rewrite the expression: \(10\times 97\times \frac{723^{13}}{10^{39}}\) - step6: Reduce the numbers: \(97\times \frac{723^{13}}{10^{38}}\) - step7: Multiply: \(\frac{97\times 723^{13}}{10^{38}}\) The amount of the element remaining after 13 minutes is approximately 14.31 grams. Therefore, to the nearest 1/10th of a gram, the amount remaining is 14.3 grams.