To find out how much Polonium-210 remains after 414 days, we need to use the concept of half-lives. Given that the half-life of Polonium-210 is 138 days, we can determine how many half-lives fit into 414 days.
First, calculate the number of half-lives in 414 days:
\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{414 \text{ days}}{138 \text{ days}} \approx 3 \]
Next, we can use the initial amount of Polonium-210 (128 grams) and determine how much remains after 3 half-lives:
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After 1 half-life (138 days), the remaining amount is: \[ 128 , \text{grams} \div 2 = 64 , \text{grams} \]
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After 2 half-lives (276 days), the remaining amount is: \[ 64 , \text{grams} \div 2 = 32 , \text{grams} \]
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After 3 half-lives (414 days), the remaining amount is: \[ 32 , \text{grams} \div 2 = 16 , \text{grams} \]
Therefore, the amount of Polonium-210 left after 414 days is 16 grams.