Question

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).

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`.

Answer 1

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]`