Answer
The half-life is $~~80~days$
Work Step by Step
We can write an expression for the decay rate:
$r = \frac{ln~2}{t_{1/2}}$
We can find the half-life:
$R = r~N$
$R = (\frac{ln~2}{t_{1/2}})~(N)$
$t_{1/2} = \frac{N~ln~2}{R}$
$t_{1/2} = \frac{(5.0\times 10^{15})~(ln~2)}{5.0\times 10^8}$
$t_{1/2} = 6.93147\times 10^6~s$
$t_{1/2} = (6.93147\times 10^6~s)[\frac{1~day}{(24)(3600~s)}]$
$t_{1/2} = 80~days$
The half-life is $~~80~days$
