Consider a window the shape of which is a rectangle of height h surmounted by a triangle having a height T that is 1.3 times the width w of the rectangle (as shown in the figure below).
If the cross-sectional area is A, determine the dimensions of the window which minimize the perimeter.
Beans
so, we have
A = hw + 1/2 h*1.3h so w = (A - 0.65w^2)/w
The perimeter is
P = w+2h+2√((w/3)^2 + (1.3w)^2)
= w + 2(A - 0.65w^2)/w + 2√((w/2)^2 + (1.3w)^2)
= 2A/w - 0.3w + √7.76 w
for minimum perimeter, dP/dw = 0, so
dP/dw = -2A/w^2 + √7.76 - 0.3
so minimum P at w = 0.897A
h = 1.11483 - 0.58305A
T = 1.3w = 1.1661A