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.
- EMLv 71 decade agoBest Answer
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)
- WendyLv 44 years ago