Let h=height where top of ladder touches wall and x be distance from wall to bottom of ladder...
tana=h/x
a=arctan(h/x) and a≤75
arctan(h/x)≤75 now taking tan of both sides :P
h/x≤tan75 now we have an ugly x value that we need to get rid of:
Using the pythagorean theorem we know:
144=x^2+h^2, x^2=144-h^2, x=√(144-h^2) now we can use this in our inequality for x
h/√(144-h^2)≤tan75
h^2/(144-h^2)≤(tan75)^2
h^2≤144(tan75)^2-h^2(tan75)^2
h^2+h^2(tan75)^2≤144(tan75)^2
h^2(1+(tan75)^2)≤144(tan75)^2
h^2≤[144(tan75)^2]/(1+(tan75)^2)
h^2≤134.353829
h≤11.5911
So he cannot have the top of the ladder 11.8 ft above the ground and not exceed a 75° angle with the ground.
I worked it the hard way just to go through the process. However we could have used a simple trig function to see that maximum height of the top of the ladder....
sin75=h/12
h=12sin75
h≈11.59 ft. That would be the maximum height given that we did not want to exceed 75° with the ground.
