# 02 - Time to dissipate 90% of certain radioactive substance

**Problem 02**

A certain radioactive substance has a half-life of 38 hour. Find how long it takes for 90% of the radioactivity to be dissipated.

**Solution 02**

$x = x_oe^{-kt}$

When t = 38 hr, x = 0.5x_{o}

$0.5x_o = x_oe^{-38k}$

$e^{-k} = 0.5^{1/38}$

Hence,

$x = x_o(0.5^{t/38})$

When 90% are dissipated, x = 0.1x_{o}

$0.1x_o = x_o(0.5^{t/38})$

$0.1 = 0.5^{t/38}$

$0.1^{38} = 0.5^t$

$t = \dfrac{38 \ln 0.1}{\ln 0.5}$

$t = 126.23 ~ \text{hrs}$ *answer*

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