When the half-life of uranium U232 92 for 74 years . What is the percentage remaining after 10 years ?
amount = a (1/2)^(t/74) where t is in years
so
when t = 10
amount = a(.5)^(10/74) = .91058.. a
so appr 91% will be left
To calculate the percentage remaining after a certain time, we can use the formula:
\( \text{{Percentage Remaining}} = \left( \frac{{\text{{Remaining Time}}}}{{\text{{Half-Life}}}} \right) \times 100 \)
In this case, we are given the half-life of uranium-232 (U232) as 74 years. To find the percentage remaining after 10 years, we first need to determine the remaining time, which is the total half-life cycles that have elapsed.
Since the given half-life is 74 years, we can calculate the number of half-life cycles in 10 years by dividing the remaining time by the half-life:
\( \text{{Number of Half-Life Cycles}} = \frac{{\text{{Remaining Time}}}}{{\text{{Half-Life}}}} = \frac{{10}}{{74}} \)
Next, we find the percentage remaining using the formula mentioned above:
\( \text{{Percentage Remaining}} = \left( \frac{{\text{{Number of Half-Life Cycles}}}}{{\text{{Total Half-Life Cycles}}}} \right) \times 100 \)
To obtain the percentage remaining, substitute the values:
\( \text{{Percentage Remaining}} = \left( \frac{{10}}{{74}} \right) \times 100 \)
Calculating the above expression:
\( \text{{Percentage Remaining}} \approx 13.51\% \)
Therefore, after 10 years, approximately 13.51% of the uranium-232 will remain.