Hello, lax600!

The problem is easier than you think ... Note that 45° angle!

The angle of elevation of a flagpole from a point level with its base is 45°.

From a point 20 ft. further away, the angle of elevation is 30°.

What is the height of the flagpole to the nearest 0.1 ft.? Code:

A *
| * *
| * *
| * *
x | * *
| * *
| * *
| 45° * 30° *
* - - - - - - - * - - - - - - - *
B x C 20 D

Since $\displaystyle \angle ACB = 45^o,\;AB = BC = x$

In right triangle $\displaystyle ABD\!:\;\;\tan30^o \:=\:\frac{x}{x+20}$

Solve for $\displaystyle x\!:\;\;x \;=\;\frac{20\tan30^o}{1 - \tan30^o}$

Since $\displaystyle \tan30^o \,=\,\frac{1}{\sqrt{3}}$, we have: .$\displaystyle x \;=\;\frac{20\cdot\frac{1}{\sqrt{3}}}{1 - \frac{1}{\sqrt{3}}} \;=\;10(\sqrt{3}+1) \;\approx\;27.3$ ft.