Answer
The appropriate aperture diameter is $~~4.2~mm$
Work Step by Step
We can see the relationship between intensity, shutter speed, and f-number:
$I \propto \frac{1}{t} \propto \frac{1}{(f-number)^2} = \frac{D^2}{f^2}$
Then:
$D^2 \propto \frac{1}{t}$
We can find the appropriate aperture diameter $D_2$:
$\frac{D_2^2}{D_1^2} = \frac{t_1}{t_2}$
$D_2^2 = (\frac{t_1}{t_2})~(D_1^2)$
$D_2 = (\sqrt{\frac{t_1}{t_2}})~(D_1)$
$D_2 = (\sqrt{\frac{1/125~s}{1/500~s}})~(3.0~mm)$
$D_2 = (\sqrt{2})~(3.0~mm)$
$D_2 = 4.2~mm$
The appropriate aperture diameter is $~~4.2~mm$