A rectangular box of length x, width y, and height z with no top is to be constructed having a volume of 32 cubic inches. Determine the dimensions that will require the least amount of material to construct the box.
A) 4 inches by 4 inches by 4 inches
B) 4 inches by 4 inches by 2 inches
C) 4 inches by 2 inches by 4 inches
D) 2 inches by 2 inches by 4 inches

Question

Quizplus · Accepted Answer

The surface area of the box (material required) is given by $S = xy + 2xz + 2yz$, and the volume $V = xyz = 32$. To minimize the surface area with a given volume, it's a problem of optimization under constraint, solvable via methods like Lagrange multipliers or by intuition and algebraic manipulation. Among the given options, B) 4 inches by 4 inches by 2 inches offers the least surface area for the given volume, as it approaches a form closer to a square base, which is more efficient in terms of surface area to volume ratio than a more elongated rectangular base or a cube when the top is open.

A Rectangular Box of Length X, Width Y, and Height