A cardboard box without a lid is to have volume 100 cubic inches, with total area of cardboard as small as possible. Find its height in inches.
A)2

B) $5 / 2$
C)4
D)5
E) $25 ^ { 1 / 3 }$
F) $25 ^ { 2 / 3 }$
G) $200 ^ { 1 / 3 }$
H) $200 ^ { 2 / 3 }$

Question

A cardboard box without a lid is to have volume 100 cubic inches, with total area of cardboard as small as possible. Find its height in inches.
A)2

B) $5 / 2$ 
C)4
D)5
E) $25 ^ { 1 / 3 }$ 
F) $25 ^ { 2 / 3 }$ 
G) $200 ^ { 1 / 3 }$ 
H) $200 ^ { 2 / 3 }$

Quizplus · Accepted Answer

The optimal dimensions for the box come from setting the derivative of the surface area function with respect to one of the dimensions equal to zero, given a fixed volume. For a box with volume $V = lwh = 100$ and surface area $S = lw + 2lh + 2wh$, using the method of Lagrange multipliers or simply understanding that for the minimum surface area the box should have all sides equal (for a box without a lid, this means the height equals the width and length), we find that the height should be the cube root of 100, which is $100^{1/3}$, making the correct answer $25^{1/3}$.

A Cardboard Box Without a Lid Is to Have Volume 5/25 / 25/2

A Cardboard Box Without a Lid Is to Have Volume $5 / 2$