Hello, Anna!

The 6 segments of the window frame shown in the diagram

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 that is 1 m wide and 3 m high.

. . . . not needed

Determine the dimensions that will maximize the area of the window. Code:

B
*
/ \
x / \ x
/ \
A * - - - * C
| x |
y | | y
| |
* - - - *
x

The total length of the framework is .$\displaystyle 4x + 2y \:=\:6 \quad\Rightarrow\quad y \:=\:3 - 2x\;\;{\color{blue}[1]}$

The area of an equilateral triangle of side $\displaystyle x$ is: .$\displaystyle \tfrac{\sqrt{3}}{4}x^2$

The area of the rectangle is: .$\displaystyle xy$

The total area is: .$\displaystyle A \;=\;\tfrac{\sqrt{3}}{4}x^2 + xy\;\;{\color{blue}[2]}$

Substitute [1] into [2]: .$\displaystyle A \;=\;\frac{\sqrt{3}}{4}x^2 + x(3-2x)$

. . and *that* is the function you must maxiimize.