To determine how much Carbon-11 will remain after 3 hours, we first need to calculate how many half-lives will occur in that time period.
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Convert hours to minutes: \[ 3 \text{ hours} = 3 \times 60 = 180 \text{ minutes} \]
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Determine the number of half-lives: The half-life of Carbon-11 is 20 minutes. Thus, the number of half-lives in 180 minutes is: \[ \text{Number of half-lives} = \frac{180 \text{ minutes}}{20 \text{ minutes/half-life}} = 9 \text{ half-lives} \]
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Calculate the remaining amount: Initially, we have 200 grams of Carbon-11. After each half-life, the amount remaining is half of the previous amount. The formula for the remaining amount after \( n \) half-lives is: \[ \text{Remaining amount} = \text{Initial amount} \times \left(\frac{1}{2}\right)^n \] where \( n \) is the number of half-lives.
Substituting the values: \[ \text{Remaining amount} = 200 \times \left(\frac{1}{2}\right)^9 \]
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Calculating \(\left(\frac{1}{2}\right)^9\): \[ \left(\frac{1}{2}\right)^9 = \frac{1}{512} \]
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Now calculate the remaining amount: \[ \text{Remaining amount} = 200 \times \frac{1}{512} = \frac{200}{512} = \frac{25}{64} \approx 0.390625 \text{ grams} \]
Thus, the amount of Carbon-11 remaining after 3 hours is approximately 0.39 grams.
Therefore, the answer is C. 0.39 grams.
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