Answer
$L = 0.354~nm$
Work Step by Step
We can write a general equation for the energy:
$E_n = \frac{n^2~h^2}{8mL^2}$
Note the following relationships:
$\sqrt{\frac{27~eV}{12~eV}} = \frac{3}{2}$
$\sqrt{\frac{48~eV}{12~eV}} = \frac{4}{2}$
Thus the $12~eV$ energy level is the $n = 2$ state.
We can find the length of the box:
$E_n = \frac{n^2~h^2}{8mL^2}$
$E_2 = \frac{(2)^2~h^2}{8mL^2}$
$L^2 = \frac{(2)^2~h^2}{8m~E_2}$
$L = \frac{(2)~h}{\sqrt{8m~E_2}}$
$L = \frac{(2)~(6.626\times 10^{-34}~J~s)}{\sqrt{(8)(9.109\times 10^{-31}~kg)~(12~eV)(1.6\times 10^{-19}~J/eV)}}$
$L = 0.354\times 10^{-9}~m$
$L = 0.354~nm$