The shadow of a tower is $40\mathrm{m}$ longer when the Sun's altitude is $30^{\circ}$ than when it is $60^{\circ}$. What is the height of the tower? 

Let the shorter shadow be $x$ and the height be $h$. For altitude $60^{\circ}$, $\tan 60^{\circ}=\frac{h}{x}$, so $h=x\sqrt{3}$. For altitude $30^{\circ}$, the shadow is $x+40$, and $\tan 30^{\circ}=\frac{h}{x+40}=\frac{1}{\sqrt{3}}$. Substituting $h=x\sqrt{3}$ gives $3x=x+40$, so $x=20$ and hence $h=20\sqrt{3}\mathrm{m}$.