Answer
The height of the antenna is $209$ ft.
Work Step by Step
1. Find the 2 missing angles in the triangle on the left
Let $A =$ the angle of the mans initial position
Let $B =$ the last unknown angle of that triangle
$A = 180 - 64$ (Angles on a horizontal line add to $180˚$
$A = 116˚$
$B + 116 + 46 = 180$ (Angles in a triangle add to $180˚$)
$B + 162 = 180$
$B = 18˚$
2. Use the sine law to solve for $x$
$\frac{x}{sin(116))} = \frac{100}{sin(18)}$
$x = \frac{100sin(116)}{sin(18)}$
by GDC / calculator
$x = \frac{89.87...}{0.309...}$
$x = 290.855...$ ft (Use exact values for the next step)
3. Use the sine law again to solve for $h$
$\frac{h}{sin(46)} = \frac{290.855...}{sin(90)}$
$h = \frac{(290.855...)(sin(46))}{1} $
$h = 209.22...$ ft
$h = 209$ ft
