QuestionMay 9, 2025

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?

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?
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?

Solution
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Answer

38.2 grams Explanation 1. Write the decay formula Use the exponential decay formula: m(t) = m_0 \cdot e^{-kt}, where m_0 is the initial mass, k is the decay rate, and t is time. 2. Calculate the decay constant k Given 27.3\% decay per minute, k = -\ln(1 - 0.273). 3. Substitute values into the formula Initial mass m_0 = 670, k = -\ln(1 - 0.273), and t = 9. Compute m(9) = 670 \cdot e^{-\ln(1 - 0.273) \cdot 9}. 4. Perform calculations k = -\ln(0.727) \approx 0.318. Then m(9) = 670 \cdot e^{-0.318 \cdot 9} \approx 670 \cdot e^{-2.862} \approx 670 \cdot 0.057 \approx 38.2 grams.

Explanation

1. Write the decay formula<br /> Use the exponential decay formula: $m(t) = m_0 \cdot e^{-kt}$, where $m_0$ is the initial mass, $k$ is the decay rate, and $t$ is time.<br /><br />2. Calculate the decay constant $k$<br /> Given $27.3\%$ decay per minute, $k = -\ln(1 - 0.273)$.<br /><br />3. Substitute values into the formula<br /> Initial mass $m_0 = 670$, $k = -\ln(1 - 0.273)$, and $t = 9$. Compute $m(9) = 670 \cdot e^{-\ln(1 - 0.273) \cdot 9}$.<br /><br />4. Perform calculations<br /> $k = -\ln(0.727) \approx 0.318$. Then $m(9) = 670 \cdot e^{-0.318 \cdot 9} \approx 670 \cdot e^{-2.862} \approx 670 \cdot 0.057 \approx 38.2$ grams.
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