The remaining mass of a radioactive substance after a certain number of half-lives can be calculated using the formula:
\[ \text{Remaining Mass} = \text{Initial Mass} \times \left(\frac{1}{2}\right)^n \]
where \( n \) is the number of half-lives.
If you start with an initial mass \( M_0 \) of cobalt-60, after three half-lives the remaining mass will be:
\[ \text{Remaining Mass} = M_0 \times \left(\frac{1}{2}\right)^3 = M_0 \times \frac{1}{8} \]
To find the specific remaining mass, we need to know the initial mass. However, since the options only provide potential final masses, we can evaluate them based on the given ratios.
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If we start with 20 g, it would be: \[ 20 , \text{g} \times \frac{1}{8} = 2.5 , \text{g} \quad (\text{Option A}) \]
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If we start with 10 g, it would result in: \[ 10 , \text{g} \times \frac{1}{8} = 1.25 , \text{g} \quad (\text{Option B}) \]
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If we start with 240 g, it would yield: \[ 240 , \text{g} \times \frac{1}{8} = 30 , \text{g} \quad (\text{Option C}) \]
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If we started with 1440 g, we would have: \[ 1440 , \text{g} \times \frac{1}{8} = 180 , \text{g} \quad (\text{Option D}) \]
Thus, without the initial mass, we cannot directly identify the answer, but each option represents the remaining mass based on different initial masses.
However, if the initial mass is not specified, we can conclude that mass options A (2.50 g) and B (1.25 g) are results of typical scenarios. Therefore, if you want just one answer without the initial mass context, the correct answer representing the transformative decay process reflected here is:
A. 2.50 g (after three half-lives with an initial mass of 20 g).
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