QuestionJune 9, 2025

An element with a mass of 600 grams decays by 29.5% per minute. To the nearest minute, how long will it be until there are 2 grams of the element remaining?

An element with a mass of 600 grams decays by 29.5% per minute. To the nearest minute, how long will it be until there are 2 grams of the element remaining?
An element with a mass of 600 grams decays by 29.5% per minute. To the nearest minute, how long will it be until there are 2 grams of the element remaining?

Solution
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Answer

15 minutes Explanation 1. Define the decay formula Use the exponential decay formula A = A_0 \cdot e^{-kt}, where A_0 is the initial amount, A is the remaining amount, k is the decay constant, and t is time. 2. Calculate the decay constant Given decay rate per minute is 29.5\%, so k = 0.295. 3. Set up the equation for remaining mass Set A = 2 grams, A_0 = 600 grams, and solve 600 \cdot e^{-0.295t} = 2. 4. Solve for time t Rearrange to e^{-0.295t} = \frac{2}{600}, then take natural logarithm: -0.295t = \ln(\frac{1}{300}). 5. Calculate t t = \frac{\ln(\frac{1}{300})}{-0.295}.

Explanation

1. Define the decay formula<br /> Use the exponential decay formula $A = A_0 \cdot e^{-kt}$, where $A_0$ is the initial amount, $A$ is the remaining amount, $k$ is the decay constant, and $t$ is time.<br />2. Calculate the decay constant<br /> Given decay rate per minute is $29.5\%$, so $k = 0.295$.<br />3. Set up the equation for remaining mass<br /> Set $A = 2$ grams, $A_0 = 600$ grams, and solve $600 \cdot e^{-0.295t} = 2$.<br />4. Solve for time $t$<br /> Rearrange to $e^{-0.295t} = \frac{2}{600}$, then take natural logarithm: $-0.295t = \ln(\frac{1}{300})$.<br />5. Calculate $t$<br /> $t = \frac{\ln(\frac{1}{300})}{-0.295}$.
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