A manufacturer designs an open-top storage crate in the shape of a rectangular prism with a square base. Exactly 1200 cm2 of thin plywood is available for the four side walls and the base (no lid). The crate must use all of the material.
(a) Let x be the length of one side of the square base (in cm) and h be the height of the crate (in cm). Derive an expression for the volume V of the crate in terms of x and h.
(b) Use the material constraint to rewrite V as a function of a single variable. Show all algebraic steps.
(c) Calculate the dimensions x and h that maximize the volume. Include explicit justification that the critical point you obtain corresponds to a maximum.
(d) Compute the maximal volume (to the nearest cubic centimeter) and interpret its meaning in the context of the crate’s design.
a. Part (a) -- derive V(x,h)
b. Part (b) -- eliminate one variable
c. Part (c) -- locate and justify the maximum
d. Part (d) -- interpret maximal volume