A 10 foot wall stands 33 feet from a building. Find the length of the shortest beam that will reach to the side of the building from the ground outside the wall. Any help will be appreciated.
Only other information that may be useful is this will create a triangle. ☺
Let θ = angle beam makes with ground
Let L = length of beam = a + b
Using diagram as a guide, we get:
sin(θ) = 10/a -----> a = 10/sinθ = 10 cscθ
cos(θ) = 33/b ----> b = 33/cosθ = 33 secθ
L = a + b
L = 10 cscθ + 33 secθ
Now L will be minimized when dL/dθ = 0
dL/dθ = -10 cotθ cscθ + 33 tanθ secθ = 0
33 tanθ secθ = 10 cotθ cscθ
33 sinθ/cosθ * 1/cosθ = 10 cosθ/sinθ * 1/sinθ
33 sin³θ = 10 cos³θ
sin³θ/cos³θ = 10/33
tan³θ = 10/33
tanθ = ∛(10/33)
θ = arctan(∛(10/33)) = 33.89°
L = 10 csc(33.89°) + 33 sec(33.89°)
L = 57.69 feet
