The six segments of the window frame (which is an equilateral triangle on top of a rectangle) are to be constructed from a piece of window framing material 6 m in length. A carpenter wants to build a frame for a rural gothic style window where triangle ABC is equilateral. The window must fit inside a space 1 m wide and 3 m wide.
Determine the dimensions that should used for the six pieces so that the maximum amount of light will be admitted. Assume no waste material for corner cuts and so on.
Let the rectangle have width x and height y. Since you specifically say "the six pieces", I presume there is a piece of frame between the triangular and rectangular pieces. The length of the frame is 4x+ 2y (4x because the base and top of the rectangle have length x and the triangular top has two sides of length x). Then the area of the rectangular part of the window is xy and the area of the triangular part is. The entire area is
. Minimize that subject to the conditions
,
, and
. (I am assuming that "The window must fit inside a space 1 m wide and 3 m wide" should have been "The window must fit inside a space 1 m wide and 3 m high".)
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