If a star has a temperature twice as hot as the Sun's and a luminosity 32 times the Sun's, what is the ratio of its radius to the Sun's radius?
The luminosity of a star is proportional to its radius squared times its temperature to the fourth power. Thus, if the star in question has a luminosity of $32$ times that of the Sun, and its temperature is twice as hot as the Sun's, we have: $$
32 = \left(\frac{R}{r}\right)^2 \left(\frac{T}{t}\right)^4 = \left(\frac{R}{1}\right)^2 \left(\frac{2}{1}\right)^4,
$$where $R$ and $r$ are the radii of the star and of the Sun respectively, and $T$ and $t$ are the temperatures of the star and the Sun respectively. Solving for $R/r$ yields $R/r = \boxed{4}$.