A box with an open top and a square base is to be constructed to contain 6,912 cubic inches. Find the dimensions that will require the minimum amount of material to construct the box.
A)The dimensions of the box using minimum material are 24 in. by 24 in. by 12 in.
B)The dimensions of the box using minimum material are 21 in. by 21 in. by 17 in.
C)The dimensions of the box using minimum material are 12 in. by 12 in. by 24 in.
D)The dimensions of the box using minimum material are 17 in. by 17 in. by 21 in.
E)The dimensions of the box using minimum material are 21 in. by 21 in. by 12 in.

Question

A box with an open top and a square base is to be constructed to contain 6,912 cubic inches. Find the dimensions that will require the minimum amount of material to construct the box. ​​
A)The dimensions of the box using minimum material are 24 in. by 24 in. by 12 in.
B)The dimensions of the box using minimum material are 21 in. by 21 in. by 17 in.
C)The dimensions of the box using minimum material are 12 in. by 12 in. by 24 in.
D)The dimensions of the box using minimum material are 17 in. by 17 in. by 21 in.
E)The dimensions of the box using minimum material are 21 in. by 21 in. by 12 in.

Quizplus · Accepted Answer

The volume of the box is given by $V = lwh = 6912$ cubic inches, where $l = w$ for a square base, and $h$ is the height. To minimize the material, we minimize the surface area $S = lw + 2lh + 2wh$. Substituting $V = l^2h = 6912$ gives $h = \frac{6912}{l^2}$. Substituting $h$ into $S$ and differentiating with respect to $l$, then solving for $l$, gives $l = w = 24$ inches. Substituting back into the volume equation gives $h = 12$ inches. Therefore, the dimensions that minimize material are 24 in. by 24 in. by 12 in.

A Box with an Open Top and a Square Base