# Differential Calculus: Optimization Problem.?

A steel girder 27 ft. long is moved horizontally along a passageway 8 ft. wide and into a corridor at right angles to the passageway. How wide must the corridor be for the girder to go around the corner?

Neglect horizontal width of the girder.

Answer is 5(5^2) [5 times square root of 5]

Just tell me how to solve this problem step by step please.

Relevance

We have to find a relative extremum for the width of the corridor (y) such that the girder can touch the inner corner and the 2 outer walls. If L is the portion of the length of the girder touching the inner corner and the outer wall in the 8 ft-passageway and θ is the angle between the girder and this same wall,

sinθ = 8/L and y = (27 - L)(cosθ)

L = 8cscθ

y = (27 - 8cscθ)(cosθ)

dy/dθ = -27sinθ + 8cscθcotθcosθ = 0

27sinθ = 8csc^2(θ)

sin^3(θ) = (8/27)

sinθ = 2/3

y = (27 - 8(3/2))sqrt(1 - (2/3)^2)

y = 15sqrt(5/9)

y = 5sqrt(5)

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