# math

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An 8 ft wall stands 27 ft from a building. Find the length of the shortest straight beam that will reach to the side of the building from the ground outside the wall.

• math -

make a sketch.
let x be the distance for the wall to where the beam touches the ground
let the height of the beam along the building be y+8 ft

by similar triangles
y/27 = 8/x
xy = 216 -----> y = 216/x

let L be the length of the beam
L^2 = (8+y)^2 + (27+x)^2
= (8 + 216/x)^2 + (27+x)^2
2L dL/dx = 2(8+216/x)(-216/x^2) + 2(27+x)
= 0 for a min of L
2(8+216/x)(-216/x^2) + 2(27+x) = 0
(8+216/x)(-216/x^2) + (27+x) = 0
-1728/x^2 - 46656/x^3 + 27+x = 0
times x^3
-1728x - 46656 + 27x^3 + x^4 = 0

x^3(x+27) - 1728(x+27) = 0
(x^3 - 1728)(x+27) = 0
x = 12 or x = -27, the last one is a "silly" answer

if x=12
L^2 = (8+18)^2 + (27+12)^2
= 2197
L = √2197 = 46.87

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