Maximizing Volume of Cylinders


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Optimal Cylinder

The optimal cylinder is a rounded cube-like shape.

 

Optimal Cylinder Formulas

Formulas for an optimal cylinder can be written to only depend on the radius by substituting \(h=2r\).

 

 

Example 3

Including the lid, the outer surface of a cylindrical rope basket has an area of \(4000\) cm2. 

A cylindrical rope basket.

Source: Basket - OZ_Media/iStock/Getty Images

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Example 4

A closed cylindrical tank is made with \(300\:000\) cm2 of sheet metal. What is the maximum capacity of the closed tank? Round your answer to the nearest litre.

Solution

Recall

Formula for volume conversion:

\(1\) L \(=1000\) cm3

For the maximum capacity or volume, the tank must be an optimal cylinder.

Let \(r\) represent the radius of the tank, \(r\gt 0\).

The height, volume, and surface area of the cylinder are

  • \(h=2r\)
  • \(V=2\pi r^3\)
  • \(SA=6\pi r^2\)

Knowing the surface area, we can substitute and solve for \(r\), then calculate the volume of the cylinder:

\[ \begin{align*} SA&=6\pi r^2 \\ 300\:000&=6\pi r^2 \\ \dfrac{300\:000}{6\pi}&=r^2 \\ r&=\sqrt{\dfrac{50\:000}{\pi}}= 126.15\ldots \end{align*}\]

Calculate the volume of the cylinder, in cm3.

\[ \begin{align*} V&=2\pi r^3 \\ &= 2\pi \left(126.15\ldots\right)^3 \\ &= 12~615~662.61\ldots \end{align*} \]

Since \(1\) L \(=1000\) cm3, the maximum capacity of the tank is approximately \(12\:616\) L.


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