Question #2ac43

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Question #2ac43

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Answer 1 · 2014-12-03T16:45:50

I believe the answer is 108 hours.

An exponential decay process can be described by the following equation:

$N (t) = N_{0} (t) \cdot {(\frac{!}{2})}^{\frac{t}{t_{\frac{1}{2}}}}$ $N(t) = N_0(t) * (!/2)^(t/t_(1/2))$ , where

$N (t)$ $N(t)$ - the initial quantity of the substance that will decay;
$N_{0} (t)$ $N_0(t)$ - the quantity that still remains and has not yet decayed after a time t;
$t_{\frac{1}{2}}$ $t_(1/2)$ - the half-life of the decaying quantity;

This being said, we know that our $t_{\frac{1}{2}}$ $t_(1/2)$ is equal to 21.6 hours, our $N (t)$ $N(t)$ is 11.25 grams, and our $N_{0} (t)$ $N_0(t)$ is equal to 360 grams.

Therefore,

$11.25 = 360 \cdot {(\frac{1}{2})}^{\frac{t}{21.6}}$ $11.25 = 360 *(1/2)^(t/21.6)$ . Now, let's say $\frac{t}{21.6}$ $t/21.6$ is equal to $y$ $y$ .
We then have

$\frac{11.25}{360} = {(\frac{1}{2})}^{y}$ $11.25/360 = (1/2)^y$

So $y = {log}_{\frac{1}{2}} (0.03125) = 5$ $y = log_(1/2)(0.03125) = 5$

Replacing this into

$\frac{t}{21.6} = 5$ $t/21.6 = 5$ we get $t = 108 h o u r s$ $t = 108 hours$

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Question #2ac43

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