Question 415189
here's how i solved it.


x = angle of elevation.
y = height of pole.


height of pole is equal to y


tan(x) = y/25


tan(2x) = y/10


the double angle tan formula is:


tan(2x) = 2tan(x) / (1 − tan^2(x))


we know that tan(2x) = y/10
we know that tan(x) = y/25


substituting in tan(2x) = 2tan(x) / (1 − tan^2(x)), we get:


y/10 = (2y/25) / (1 - (y/25)^2)


simplify to get:


y/10 = (2y/25) / (1 - (y^2/625))


multiply both sides of the equation by (1 - (y^2/625) and divide both sides of the equation by (y/10) to get:


1 - (y^2/625) = (2y/25) / (y/10)


this is equivalent to:


1 - (y^2/625) = (2y/25) * (10/y)


simplify to get:


1 - (y^2/625) = 20/25


add (y^2/625) to both sides of the equation and subtract 20/25 from both sides of the equation to get:


1 - (20/25) = y^2/625


simplify to get:


5/25 = y^2/625


multiply both sides of the equation by 625 to get:


(625*5)/25 = y^2


simplify to get:


125 = y^2


take square root of both sides of the equation to get:


y = +/- 11.18033989


you have tan(x) = 11.18033989/25 = .447213596


arctan (.447213596) = 24.09484255 degrees = x.


you also have tan(2x) = 11.18033989/10 = 1.118033989


arctan(1.118033989) = 48.1896851 degrees = 2x


the results check out ok.


y = 11.18033989 feet which is the height of the pole.


formulas we started with are:


y = 25 * tan(x) and y = 10 * tan(2x)


our angles became:


x = 24.09484255 degrees
2x = 48.1896851 degrees


tan(x) = .447213596
tan(2x) = 1.118033989


y = 25*tan(x) = 25 * .447213596 = 11.18033989
y = 10*tan(2x) = 10 * 1.118033989 = 11.18033989


height of the pole is 11.18033989 feet.