From a hot air balloon, the angle between a radio antenna straight below and the base of the library downtown is 63°, as shown below. If the distance between the radio antenna and the library is 24 miles, how many miles high is the balloon?
+33663 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
+33663 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
