Answer
The flagpole's height is approximately $388in$
Work Step by Step
Let's imagine a triangle on this image. It will be equilateral triangle (As shown on the image).
Let's also consider that the pole is $h$.
Each side will be equal to $(h+5)ft$. According to one of the characteristics of a right angled triangle, the leg (side) opposite to $30°$ angle equals to half of hypotenuse. Which means that it is $\frac{h+5}{2}ft$
Now we can easily find flagpole height using the Pythagoras Theorem (Let's consider flagpole height as $h$) :
$(\frac{h+5}{2})^2+h^2=(h+5)^2$
$\frac{h^2+10h+25}{4}+h^2=h^2+10h+25$
$h^2+10h+25=40h+100$
$h^2-30h-75=0$
$D = b^2-4ac = (-30)^2 - 4\times(-75)=900+300=1200$
$h_1 = \frac{-b-\sqrt{D}}{2a}=\frac{30-\sqrt{1200}}{2}$; This is a negative number and a length can't be negative number, so we can simply cross out this result.
$h_2 = \frac{-b+\sqrt{D}}{2a}=\frac{30+\sqrt{1200}}{2}=15+10\sqrt3\approx32.32$
The flagpole is $32.32ft\approx387.84\approx388in$
