It shall be a square beam of side equal to D/2.What have you tried?
Do you recognize that this problem boils down to maximizing the area of a rectangle inscribed within a circle?
As always, what is your constraint? What is your objective function?
Hint: look at the diagonal of the rectangle and how it relates to the diameter of the circle. And then how do the sides of the rectangle relate to its diagonal?
Ok I got y = x = D/sqrt(2)Here is a cross-section:
View attachment 2086
Our objective function is the area of the rectangle:
Subject to the constraint (by Pythagoras):
So, solve the constraint for either variable, and substitute for that variable into the objective function so that you only have one variable, and then maximize.