Finding window dimensions

Question

A Norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. Find the exact dimensions (in ft) of a Norman window of maximum area if the total perimeter is 10 feet.

Answer 1

Rectangular window: Width = w, Height = h

Semicircular window radius = w/2

Perimeter P = w + 2h + (1/2)[2π(w/2)] = w + 2h + π(w/2) = w(1 + π/2) + 2h = w((π + 2)/2) + 2h

==> P₀ = 10 ==> P₀ = w((π + 2)/2) + 2h ==> 2P₀ = w(π + 2) + 4h ==> h = P₀/2 - w(π + 2)/4

Area A = hw + (1/2)π(w/2)² = hw + (π/8)w² = w(P₀/2 - w(π + 2)/4) + (π/8)w² = (P₀/2)w + w²[(π/8) - (π + 2)/4]

A = (P₀/2)w + w²[π/8 - π/4 - 1/2] = (P₀/2)w - (π/8 + 1/2)w²

Need dA/dw = 0 = (P₀/2) - 2(π/8 + 1/2)w ==> (π/4 + 1)w = (P₀/2) ==> (1/4)(π + 4)w = (P₀/2) ==> w = 2P₀ / (π + 4)

Back to height h = P₀/2 - w(π + 2)/4 = 2P₀/4 - w(π + 2)/4 = (1/4)(2P₀ - (2P₀ / (π + 4))(π + 2) )

h = (1/2)(1/(π + 4))(2P₀(π + 4) - (2P₀)(π + 2) ) = (1/2)(1/(π + 4))(P₀(π + 4) - P₀(π + 2) ) = (1/2)(1/(π + 4))2P₀

h = (1/(π + 4))P₀ ==> h = P₀ / (π + 4)

Ans. Rectangular Height h = P₀ / (π + 4), Rectangular Width w = 2P₀ / (π + 4) ==> w = 2h

Answer 2

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