Question

an element with mass 670 grams decays by 21.8% per minute. how much of the element is remaining after 15 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, \( P = 670 \) grams, \( r = 0.218 \) (since 21.8% = 0.218), and \( t = 15 \) minutes.

Step2: Substitute the values into the formula

Substitute \( P = 670 \), \( r = 0.218 \), and \( t = 15 \) into \( A = P(1 - r)^t \):
\( A = 670(1 - 0.218)^{15} \)
First, calculate \( 1 - 0.218 = 0.782 \).
Then, calculate \( 0.782^{15} \). Using a calculator, \( 0.782^{15} \approx 0.0374 \).

Step3: Calculate the final amount

Multiply \( 670 \) by \( 0.0374 \):
\( A = 670 \times 0.0374 \approx 25.058 \)

Step4: Round to the nearest tenth

Rounding \( 25.058 \) to the nearest tenth gives \( 25.1 \).

Answer:

\( 25.1 \) grams