what is the maximum volume of a square pyramid that can fit inside a cube with a side length of 18 cm?
To find the maximum volume of a square pyramid that can fit inside a cube with a side length of 18 cm, we want to find the largest possible square pyramid that can fit inside the cube.
If we draw the cross-section of the pyramid and the cube, we can see that the pyramid's base is a square with side length "x", and the pyramid's height is also "x", because it is perpendicular to the base.
[asy]
draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);
draw((0.5,0)--(0.5,0.5));
label("x", (0,0.5), W);
label("x", (0.5,1), N);
label("x", (1,0.5), E);
label("x", (0.5,0), S);
draw((1.5,0)--(3,0)--(3,1.5)--(1.5,1.5)--cycle);
draw((1.75,0)--(1.75,1.75));
draw((1.5,1.5)--(3,1.5));
draw((1.5,1.5)--(1.75,1.75));
label("18", (1.5,-0.2), S);
label("18", (3,0.6), E);
label("18", (2.7,1.5), N);
label("$x$", (1.5,-0.2)--(1.5,1.5), W);
[/asy]
We know that the square pyramid needs to fit completely inside the cube, so the maximum possible value of "x" is 18 cm. Therefore, the maximum volume of the square pyramid is:
$\frac{1}{3} x^2 h = \frac{1}{3} (18 \text{ cm})^2 (18 \text{ cm}) = \boxed{11,\!376 \text{ cm}^3}$.