Question

1. Joshua has a ladder that is 12 ft long. He will lean the ladder against a vertical wall. For safety reasons, he wants the angle the ladder makes with the ground to be no greater than 75°. Is it possible for Joshua to lean the ladder against the wall so that the top of the ladder is at least 11.8 ft above the ground?
Show your work and draw a diagram to support your answer.

Answer 1

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.