Length of a beam?

Question

An 8 ft wall stand 27 ft from a building. Find the length of a beam that will reach the side of the building from the outside of the wall.

forgot to type it. but yeah, im looking for the shortest length of the beam.

Answer 1

You want the beam of minimum length, not just any beam, right? (Otherwise, for example, a beam of length 35√2 ft. will work; it will make a 45° angle with the ground.)

See figure: http://i6.tinypic.com/82sjb6g.gif

Method 1: the length L is L1 + L2. Show that L1 = 8 csc(θ) and L2 = 27 sec(θ) so

L = 8 csc(θ) + 27 sec(θ)

Minimize L with respect to θ

(You should find that tan(θ) = 2/3.)

Method 2: By similar triangles, 8/x = y/27, so y = 216/x.

L = √[(27 + x)² + (8 + y)²] = √[(27 + x)² + (8 + 216/x)²]

Minimize L with respect to x. You should get x=12.

With either method, you will find that the minimum L is 13√13 ft.