Answer
$D=171m$
Work Step by Step
The formula relating distance from a source and intensity is $$I=\frac{P}{A}=\frac{P}{4\pi R^2}$$ The original intensity, $I_o$, can be written as $$I_o=\frac{P}{4\pi D^2}$$ The new intensity, $I_f$, has a radius of $D-50$, leaving an intensity of $$I_f=\frac{P}{4\pi (D-50)^2}$$ Note that $I_f=2I_o$. This means that $$\frac{P}{4\pi(D-50)^2}=\frac{2P}{4\pi D^2}$$ Dividing each side by $\frac{P}{4\pi}$ leaves $$\frac{1}{(D-50)^2}=\frac{2}{D^2}$$ Cross multiplying leaves $$D^2=2(D-50)^2$$ Expanding terms leaves $$D^2=2D^2-200D+5000$$ Solving for D gives $$D^2-200D+5000=0$$ $$D=171m$$
