Answer
The height of the tower is 87.3 ft
Work Step by Step
Let $d$ be the horizontal distance from the building to the tower. Let $x$ be the height of the tower.
We can use the law of sines and the angle at the base of the building to find an expression for $d$:
$\frac{d}{sin(90^{\circ}-29^{\circ}30')} = \frac{x}{sin(29^{\circ}30')}$
$d = \frac{x~sin(60^{\circ}30')}{sin(29^{\circ}30')}$
We can use the law of sines and the angle on the roof of the building to find an expression for $d$:
$\frac{d}{sin(90^{\circ}-15^{\circ}20')} = \frac{x-45}{sin(15^{\circ}20')}$
$d = \frac{(x-45)~sin(74^{\circ}40')}{sin(15^{\circ}20')}$
We can equate the two expressions for $d$ and solve for $x$:
$ \frac{x~sin(60^{\circ}30')}{sin(29^{\circ}30')} = \frac{(x-45)~sin(74^{\circ}40')}{sin(15^{\circ}20')}$
$x~sin(60^{\circ}30')~sin(15^{\circ}20') = (x-45)~sin(74^{\circ}40')~sin(29^{\circ}30')$
$0.2301469~x = 0.474896~(x-45)$
$0.24475~x = 21.37$
$x = \frac{21.37}{0.24475}$
$x = 87.3~ft$
The height of the tower is 87.3 ft
