A tent is constructed in the shape of a pyramid with square base and four faces; each face is

in the shape of an isosceles triangle. The tent is made by cutting away a triangle from each side of a

6 ft x 6 ft silicon-coated polyester and bending up the the resulting triangles to form the walls of

the tent. A steel pole, perpendicular to the base, is placed in the center of the tent for support.

(PICTURE OF FIGURE ATTACHED BELOW)

a) Show that the function to be optimized is V(x) = $\frac{1}{3}{x}^2\sqrt{18-3\sqrt{2}x}$

where x represents the side length of the square base. Provide an appropriate domain for V .

Fully justify every step of the derivation of the function.

b) Find the maximum volume the tent can have. Fully justify your answer.