Question

an element with mass 670 grams decays by 27.3% per minute. how much of the element is remaining after 9 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, \( r = 0.273 \) (since 27.3% = 0.273), and \( t = 9 \) minutes.

Step2: Substitute the values into the formula

Substitute \( P = 670 \), \( r = 0.273 \), and \( t = 9 \) into \( A = P(1 - r)^t \):
\( A = 670(1 - 0.273)^9 \)
First, calculate \( 1 - 0.273 = 0.727 \).
Then, calculate \( 0.727^9 \). Let's compute this step by step:
\( 0.727^2 = 0.727\times0.727 \approx 0.5285 \)
\( 0.727^4=(0.727^2)^2\approx0.5285^2\approx0.2793 \)
\( 0.727^8=(0.727^4)^2\approx0.2793^2\approx0.0780 \)
\( 0.727^9 = 0.727^8\times0.727\approx0.0780\times0.727\approx0.0567 \)
Now, multiply by 670: \( A = 670\times0.0567 \approx 670\times0.0567 \)
\( 670\times0.05 = 33.5 \), \( 670\times0.0067 = 4.489 \)
Adding them together: \( 33.5 + 4.489 = 37.989 \approx 38.0 \) (using a calculator for more accurate \( 0.727^9 \): \( 0.727^9\approx0.05667 \), then \( 670\times0.05667\approx37.97 \approx 38.0 \))

Answer:

38.0