QuestionApril 26, 2025

A radioactive isotope has a half-life of 13 hours. Find the amount of the isotope left from a 400 -milligram sample after 52 hours. If necessary, round your answer to the nearest thousandth. 12.5 mg 25 mg 7.692 mg 0.049 mg

A radioactive isotope has a half-life of 13 hours. Find the amount of the isotope left from a 400 -milligram sample after 52 hours. If necessary, round your answer to the nearest thousandth. 12.5 mg 25 mg 7.692 mg 0.049 mg
A radioactive isotope has a half-life of 13 hours. Find the amount of the isotope left from a 400 -milligram sample after 52 hours. If necessary, round your answer to
the nearest thousandth.
12.5 mg
25 mg
7.692 mg
0.049 mg

Solution
4.1(34 votes)

Answer

25 mg Explanation 1. Use the half-life decay formula The formula for radioactive decay is A = A_0 \cdot (0.5)^{t/T}, where A is the remaining amount, A_0 is the initial amount, t is the elapsed time, and T is the half-life. 2. Substitute values into the formula A_0 = 400, t = 52, and T = 13. Substituting these values gives: A = 400 \cdot (0.5)^{52/13}. 3. Simplify the exponent 52/13 = 4, so: A = 400 \cdot (0.5)^4. 4. Calculate the power of 0.5 (0.5)^4 = 0.0625, so: A = 400 \cdot 0.0625. 5. Multiply to find the final amount A = 25 milligrams.

Explanation

1. Use the half-life decay formula<br /> The formula for radioactive decay is $A = A_0 \cdot (0.5)^{t/T}$, where $A$ is the remaining amount, $A_0$ is the initial amount, $t$ is the elapsed time, and $T$ is the half-life.<br /><br />2. Substitute values into the formula<br /> $A_0 = 400$, $t = 52$, and $T = 13$. Substituting these values gives:<br />$A = 400 \cdot (0.5)^{52/13}$.<br /><br />3. Simplify the exponent<br /> $52/13 = 4$, so:<br />$A = 400 \cdot (0.5)^4$.<br /><br />4. Calculate the power of 0.5<br /> $(0.5)^4 = 0.0625$, so:<br />$A = 400 \cdot 0.0625$.<br /><br />5. Multiply to find the final amount<br /> $A = 25$ milligrams.
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