# Thread: Optimization , pls help!

1. ## Optimization , pls help!

A memorial window is in the form of a rectangle topped by an equilateral triangle with each side equal to the width of the rectangle. If the perimeter of the window is 900 cm, what is its maximum area?

2. Originally Posted by robertosavin
A memorial window is in the form of a rectangle topped by an equilateral triangle with each side equal to the width of the rectangle. If the perimeter of the window is 900 cm, what is its maximum area?
Hi roberto,

If we denote x as the width of the rectangle and the side of the triangle,
then perimeter is

3x+2y=900

the area of the window is the rectangle area plus the triangle area

$\displaystyle 0.5x^2sin60^o+xy$

To maximise, we need to differentiate wrt a single variable,

so write y in terms of x

$\displaystyle 3x+2y=900\ \Rightarrow\ y=\frac{900-3x}{2}$

now substitute that y into the area equation,
differentiate it wrt x,
set the result =0
find the x that causes it to be zero and use it to calculate max area.