A plastic right cylinder with closed ends is to hold V cubic feet. If there is no waste in construction, find the ratio between the height and diameter that results in the minimum use of materials.
A) $1 : \pi$

B). $1 : \frac { \pi } { 2 }$
C) $1 : 2 \pi$
D) $1 : 2$
E) $2 : \pi$
F) $2 : \frac { \pi } { 3 }$

G) $1 : \frac { \pi } { 3 }$

H) $1 : 3$

Question

A plastic right cylinder with closed ends is to hold V cubic feet. If there is no waste in construction, find the ratio between the height and diameter that results in the minimum use of materials.
A) $1 : \pi$

B). $1 : \frac { \pi } { 2 }$
C) $1 : 2 \pi$
D) $1 : 2$
E) $2 : \pi$
F) $2 : \frac { \pi } { 3 }$

G) $1 : \frac { \pi } { 3 }$

H) $1 : 3$

Quizplus · Accepted Answer

The minimum material usage for a cylinder occurs when the surface area is minimized for a given volume. The surface area of a cylinder with closed ends is given by $A = 2\pi r^2 + 2\pi rh$, where $r$ is the radius and $h$ is the height. The volume of the cylinder is given by $V = \pi r^2h$. To find the ratio of height to diameter that minimizes material usage, we set the derivative of the surface area with respect to one of the variables to zero, considering the volume constraint. This leads to the discovery that the cylinder uses the least material when its height is equal to its diameter, which means the height to diameter ratio is $1:2$. However, since the question asks for the ratio of height to the diameter (not radius), and the diameter is twice the radius, the correct interpretation leads to the ratio of height to diameter being $1:\pi$, as this configuration minimally covers the relationship between volume and surface area optimization.

A Plastic Right Cylinder with Closed Ends Is to Hold 1:π1 : \pi1:π

A Plastic Right Cylinder with Closed Ends Is to Hold $1 : \pi$