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.

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.

No because the angle will be greater than 75 degrees