What's the value of h which minimize the perimeter?

Consider a window the shape of which is a rectangle of height h surmounted by a triangle having a height T that is 0.9 times the width w of the rectangle (as shown in the figure below).

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

I found the value of w but couldn't find the value of h.

1 Answer
Nov 21, 2017

See below.

Explanation:

Cross sectional area

A = h * w +1/2T*w = h*w+1/2*0.9*w^2

Perimeter

P = w + 2*h + 2 sqrt(T^2+(w/2)^2) = w + 2*h + 2 w sqrt(0.9^2+(1/2)^2) = 2*h + (2sqrt(0.9^2+(1/2)^2) + 1)w

so calling c_0 = 0.9/2, c_1 = 2, c_2 = (2sqrt(0.9^2+(1/2)^2) + 1)

we have

find

min P = c_1h+c_2w

subjected to

A = h*w+c_0 w^2

substituting h = (A-c_0 w^2)/w into P we get

P = c_1 w + (c_1 (A - c_0 w^2))/w

and

(dP)/(dw) = -2 c_0 c_1 + c_2 - (c_2 (A - c_0 w^2))/w^2 = 0

giving

w_0 = (sqrt[c_1A])/sqrt[c_2-c_0 c_1] = 0.962445 sqrt[A]

and consequently

h_0 = (A-c_0 w_0^2)/w_0 = 2sqrt(A c_1(c_2-c_0c_1)) = 0.60592 sqrt[A]