** lilit_movsisyan asks**:

In other words, what container (spherical, cylindrical, conical, etc.) could hold 12 ounces of liquid and have the LEAST surface area? And what are the dimensions (height, radius, side lengths, etc) of this container?

Note: 12 ounces = 354 cm^3 (this is the conversion factor, as it made it a lot easier)

So far, I have a cylinder as the most efficient container.

12 ounces = 354.882 cm^3***

I’m supposed to present this with proof involving calculus (application of derivatives/optimization,etc). And I’ve tried a cylinder and a cone. The cylinder had less between those two.

Thanks for the answers guys.

My calculated MINIMUM surface areas are:

Cylinder: 277.490 cm²

Cone: 305.409 cm²

Sphere: 242.401 cm²

**Best answer:**

*Answer by Big Daddy*

Have you checked both cubes and spheres?

A sphere has the lowest surface area to volume ratio of all shapes.

A sphere is the three-dimensional shape with the largest volume to surface area ratio. You can find some proofs of this on-line if you search around. So use a spherical container. Set (4/3)pi r^3 equal to the volume in cubic centimeters, then solve for r to find the necessary radius in centimeters. The surface area will be 4 pi r^2. If you check, it should be lower than that of the cylinder.