Question #ab040

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Question #ab040

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Answer 1 · 2015-09-11T23:26:07

The answer is (B) $11520 years$ $"11520 years"$

Explanation:

A radioactive isotope's half-life tells you the time needed for an initial sample of said isotope to be halved.

More specifically, an initial sample of a radioactive element will be halved for every half-life that passes. If you start at $t = 0$ $t=0$ and have $t_{1/2}$ $t_"1/2"$ as the isotope's half-life, then an initial sample will change like this

$100 % \to t = 0$ $100% -> t = 0$

$50 % \to t = t_{1/2}$ $50% -> t = t_"1/2"$

$25 % \to t = 2 \cdot t_{1/2}$ $25% -> t = 2 * t_"1/2"$

$12.5 % \to t = 3 \cdot t_{1/2}$ $12.5% -> t = 3 * t_"1/2"$

$6.25 % \to t = 4 \cdot t_{1/2}$ $6.25% -> t = 4 * t_"1/2"$

This is equivalent to saying that

$what you have = \frac{what you started with}{2^{n}}$ $"what you have" = "what you started with"/2^n" "$ , where

$n$ $n$ - the number of half-lives that passed.

In your case, you start with a sample of 100 mg. In order for the sample to be reduced to 25 mg, you need to have

$100 mg \to t = 0$ $"100 mg" -> t = 0$

$50 mg \to t = t_{1/2}$ $"50 mg" -> t = t_"1/2"$

$25 mg \to t = 2 \cdot t_{1/2}$ $"25 mg" -> t = 2 * t_"1/2"$

Your remaining sample is now a quarter the size it was in the beginning, which can only mean that two half-lives have passed

$25 mg = \frac{100 mg}{2^{n}}$ $"25 mg" = "100 mg"/2^n$

$2^n = (100color(red)(cancel(color(black)("mg"))))/(25color(red)(cancel(color(black)("mg")))) = 4 implies n = color(green)(2)$

The time that passed is thus equal to

$t = 2 * t_"1/2" = 2 * "5760 years" = color(green)("11520 years")$

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Question #ab040

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