What Is the Maximum Area of a Gothic Window With Given Constraints?

[roman15]

Homework Statement

a gothic window it to be built with 6 segments that total 6m in length. The window must fit inside an area that is 1m wide and 3 meters tall. the triangle on top must be equilateral. What is the maximum area of the window.

Homework Equations

The Attempt at a Solution

so i made each of the sides of the trianle x and the width at the bottom of the rectangle x, then the sides of the rectangle equal to y. so the perimeter is 6=4x+2y
and the area A=1/2xh + xy
the height of the triangle would be 3-y wouldn't it? because then i plugged that in for h, and then isolated for y in the perimeter equation and plugged in y where it was necessary but i didnt get the right answer in the end after i got the derivative

 

[Dick]

No, you don't know h=3-y. Given your perimeter constraint the maximum area window might not touch all sides of the 1m by 3m area. All you really know is that x<=1 and y+h<=3. What is true that h is related to x just because x is the side of an equilateral triangle and h is the height.

 

[rekshaw]

I'm assuming your shape is like this?

[u]/\[/u]
|_|

Since the top has to be an equilateral triangle you know that 3 segments have to be equal (if the base of the triangle counts as a segment). Then you have 3 segments left over to play with (the sides below the triangle and the base of the window)