Question

A rectangular beam is to be cut from the log into a circular cross section. If the strength of the beam is proportional to the width and the square of the depth, find the dimensions that will give the strongest beam

Answer 1

I assume we can center the beam in the circle. So, let
2x = width
2y = depth
That means that x^2+y^2 = r^2, the radius of the log
If we let the strength be z, then we know that
z = kxy^2 = kx(r^2-x^2) = kr^2x - kx^3
dz/dx = k(r^2 - 3x^2)
Now just find x where dz/dx=0
You can see that z will be a maximum.