# Help :(( Exponential Decay

a radio active element follows the law of exponential change and has a half-life of 38 hours. how long will it take for 90% of the radioactivity of the element to be dissipated ?

### Re: Help :((

Following exponential change
$x = x_o e^{-kt}$

After 38 hours, 50% is dissipated, hence 50% retained
$0.5x_o = x_o e^{-38k}$

$0.5 = e^{-38k}$

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

Thus,
$x = x_o 0.5^{t/38}$

90% dissipated means 10% retained, hence, x = 0.10xo
$0.10x_o = x_o 0.5^{t/38}$

$0.10 = 0.5^{t/38}$

$t = 126.23 ~ \text{hrs}$

### Re: Help :(( Exponential Decay

You can also use exponential STAT in your calculator:

CASIO fx-991ES PLUS: [MODE]   [3:Stat]   [5:e^X]

 X (time) Y (% retained) 1 0 1 2 38 0.5

[AC]
For 10% retained:
X (time) = 0.1x-caret = 126.23 hours

x-caret can be found at [Shift]   [1:Stat]   [5:Reg]   [4:x-caret]

• Mathematics inside the configured delimiters is rendered by MathJax. The default math delimiters are $$...$$ and $...$ for displayed mathematics, and $...$ and $...$ for in-line mathematics.