The level of radioactivity on the site of a nuclear explosion is decaying exponentially. The level measured in 1970 was found to be 0.7 times the level measured in 1960. What is the half-life. (the amount of years)

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- Nov 6th 2007, 09:52 PMmath34aradioactive
The level of radioactivity on the site of a nuclear explosion is decaying exponentially. The level measured in 1970 was found to be 0.7 times the level measured in 1960. What is the half-life. (the amount of years)

- Nov 6th 2007, 10:59 PMjanvdl
Radioactivity in 1970 = $\displaystyle x$

Radioactivity in 1960 = $\displaystyle y$

But $\displaystyle x = 0,7y$

$\displaystyle 0,7y = y(1 - \frac{r}{100} ) ^{10}$

$\displaystyle 0,7 = (1 - \frac{r}{100} ) ^{10}$ [Divide by y]

$\displaystyle (1 - \frac{r}{100} ) = \sqrt[10]{0,7}$

$\displaystyle (1 - \frac{r}{100} ) = 0,9649$

$\displaystyle \frac{r}{100} = 0,0350$

$\displaystyle r = 3,503$

I think this is the rate at which it is decaying. I'll spend a little more time on it... :D

EDIT: Okay i got it! We now have the rate, so we want to know when $\displaystyle x = 0,5y$

$\displaystyle 0,5y = y(1 - \frac{3,503}{100}) ^ {n} $

$\displaystyle 0,5 = (1 - \frac{3,503}{100}) ^ {n} $

$\displaystyle n = log_{ 1 - \frac{3,503}{100} }{0,5} $

$\displaystyle n = 19,43 \ years $ :D