Consider a window the shape of which is a rectangle of height h surmounted by a triangle having a height T that is 0.5 times the width w of the rectangle (as shown in the figure below).

Question

If the cross-sectional area is A, determine the dimensions of the window which minimize the perimeter.

h =
w =

Answer 1

a little check will show that
h = A/w - w/4

So, the perimeter is (assuming the triangle is isosceles)

p = w+2h+w√2 = w(1+√2) + 2(A/w - w/4)
= (1/2 + √2)w + 2A/w

dp/dw = (1/2 + √2) - 2A/w^2
dp/dw = 0 when
w = 2√(A/(2+√2))
h = (1+√2)/2 √(A/(2+√2))