Question

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 10th of a gram?

Explanation:

Step1: Identify the decay formula

The formula for exponential decay is \( A = P(1 - r)^t \), where \( A \) is the final amount, \( P \) is the initial amount, \( r \) is the rate of decay (as a decimal), and \( t \) is the time.

Step2: Convert the decay rate to a decimal

The decay rate is \( 27.7\% = 0.277 \). So the remaining rate per minute is \( 1 - 0.277 = 0.723 \).

Step3: Substitute the values into the formula

We have \( P = 970 \), \( r = 0.277 \) (so the base is \( 0.723 \)), and \( t = 13 \). Plugging into the formula: \( A = 970 \times (0.723)^{13} \).

Step4: Calculate the value

First, calculate \( (0.723)^{13} \). Using a calculator, \( (0.723)^{13}\approx0.01023 \). Then multiply by 970: \( A = 970\times0.01023\approx9.9231 \). Rounding to the nearest tenth gives \( 9.9 \).

Answer:

\( 9.9 \) grams