Verify if I did it right

[SKG_Scintill]

Ever so often I get bored during classes and decide to do some math.
I haven't done this kind of math since high school (only 2 years ago, but it's good to keep your mind fresh), so I have the feeling I forget stuff every once in a while. My teachers aren't of much help, they know seemingly less maths than I do.

Now, for the question at hand:
We start with a square pyramid(ABCDE) with base dimensions of 10 cm by 10 cm and a height of 20 cm.
We want to know what the largest possible volume of an inscribed cuboid is (inside the pyramid obviously)

Pyramid(ABCDE) = 10 cm x 10 cm x 20 cm

f(x) = 4x * (10 - 2x) * x
     = 4x * (10x - 2x^2)
     = 40x^2 - 8x^3 = -8x^3 + 40x^2
f'(x) = -24x^2 + 80x

-24x^2 + 80x = 0
x(-24x + 80) = 0
x = 0 v -24x + 80 = 0
	24x = 80
	x = 80/24 = 3 + 1/3

f(3 + 1/3) = 4 * (3 + 1/3) * (10 - 2 * (3 + 1/3)) * (3 + 1/3)
	   = (13 + 1/3) * (3 + 1/3) * (3 + 1/3)
	   = 148 + 4/27 cm^3

Cube(ABCDEFGH) ≈ 3.33 cm x 3.33 cm x 13.33 cm
Volume(ABCDEFGH) ≈ 148.148 cm^3

This seemed a bit... too logical... did I do it right?

Edit: Gonna make a drawing with paint soon

[Arkuski]

I'm getting twice the answer that you got. I'll walk you through the thought process.

The pyramid has a 10x10 square base and a height of 20. We will observe a half cross section of the pyramid in the form of a right triangle with height 20 and base 5. The hypotenuse of this triangle can be described by the function y=20-4x. Suppose we want to evaluate the volume of the cuboid at point x. Then we find its height is 20-4x, its width is 2x, and its length is also 2x due to the square nature of the pyramid.

Now that we have these values it's time to do some calculus. We want to maximize V=(20-4x)(2x)^2:

V = 80x^2 - 16x^3
dV/dx = 160x - 48x^2
48x^2 = 160x
48x = 160
x = 160 / 48
x = 10 / 3

Now that we have our value for x, we substitute it back into the equation:

V = (20 - 4 * (10 / 3)) * (2 * (10 / 3))^2
V = (20 - (40 / 3)) * (20 / 3)^2
V = (20 / 3) * (20 / 3)^2
V = (20 / 3)^3
V = 8000 / 27
V = 296.296...