(Board Across Hallways) In a building, two intersecting halls with widths w1 = 9 feet an...

(Board Across Hallways) In a building, two intersecting halls with widths w₁ = 9 feet and w₂ = 7 feet meet at an angle α = 125^?, as shown:

Assuming a two-dimensional situation, what is the longest board that can negotiate the turn? Ignore the thickness of the board. The relationship between the angles θ and the length of the board l = l₁ + l₂ is l1 = w₁ csc(β), l₂ = w₂ csc(γ ), β = π − α − γ and l = w₁ csc(π − α − γ ) + w₂ csc(γ ). The maximum length of the board that can make the turn is found by minimizing l as a function of γ . Taking the derivative and setting dl/dγ = 0, we obtain

w₁ cot(π−α−γ ) csc(π−α−γ )−w₂ cot(γ ) csc(γ ) = 0

Substitute in the known values and numerically solve the nonlinear equation.