please help!!

tan(64°) is given by the distance from radio mast to library, divided by height of balloon (h), or:

tan(64°)=24/h

Rearrange to get h = 24/tan(64°)

$${\mathtt{h}} = {\frac{{\mathtt{24}}}{\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}{\left({\mathtt{64}}^\circ\right)}}} \Rightarrow {\mathtt{h}} = {\mathtt{11.705\: \!582\: \!125\: \!582\: \!365\: \!3}}$$  

h ≈ 11.7 miles

 Edit:  I should have used 63° not 64° in the above!!  

         Thanks for pointing this out Chris.

$${\mathtt{h}} = {\frac{{\mathtt{24}}}{\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}{\left({\mathtt{63}}^\circ\right)}}} \Rightarrow {\mathtt{h}} = {\mathtt{12.228\: \!610\: \!787\: \!867\: \!229\: \!7}}$$

h ≈ 12.2 miles