Question

an element with mass 570 grams decays by 26.9% per minute. how much of the element is remaining after 14 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 (in decimal), and \( t \) is the time.
Here, \( P = 570 \) grams, \( r = 26.9\% = 0.269 \), and \( t = 14 \) minutes.

Step2: Substitute the values into the formula

Substitute \( P = 570 \), \( r = 0.269 \), and \( t = 14 \) into the formula:
\( A = 570(1 - 0.269)^{14} \)
First, calculate \( 1 - 0.269 = 0.731 \).
Then, calculate \( 0.731^{14} \). Using a calculator, \( 0.731^{14} \approx 0.0243 \).
Now, multiply by 570: \( A = 570 \times 0.0243 \approx 13.851 \).

Step3: Round to the nearest tenth

Rounding \( 13.851 \) to the nearest tenth gives \( 13.9 \).

Answer:

13.9