Question

find the most economical dimensions for a closed cylindrical can containing a quart. ANS is DIAMETER= height explain clearly

Answer 1

It sounds like "most economical dimensions" means you want to use the least amount of material for the can so that it still contains a quart. In other words, you're minimizing the surface area of the can under the restraint of a quart of volume.

The surface area of the can is given by
\[A=2\pi rh+2\pi r^2\]
and its volume is
\[V=1\text{ quart}=\pi r^2h\]
You want to show that \(A\) is minimized when \(h=d=2r\).