The math help thread

Now this one is considerably trickier.

To determine the new dimensions of the circumscribed rectangle based on the original rectangle's dimensions L and W, and the variable θ, we'll apply geometry and trigonometry.

Code:

sin(θ) = o/h
cos(θ) = a/h
For both cases, h will either be L or W.

X = sin(θ)*W + cos(θ)*L; new rectangle's length
Y = sin(θ)*L + cos(θ)*W; new rectangle's width

V = X*Y
  = sin(θ)^2*L*W + cos(θ)^2*L*W + sin(θ)*cos(θ)*(W^2 + L^2)
  = L*W + sin(θ)*cos(θ)*(W^2 + L^2)

We'll do the same trick as with the cone problem: find the derivative of the volume function based off of θ.

Code:

V' = (-sin(θ)^2 + cos(θ)^2)*(W^2 + L^2)

If θ = Π/4 = 45°, V' = 0.

[color=Red]V = L*W + 1/2(W^2 + L^2)[/color]

where did you get the LW?
This is how I did it.