Best Cylinder

Calculus Level 2

A cylinder made of an iron plate can contain $2000 \text{ cm}^3$ of liquid.

What is the radius of the cylinder (in cm) that minimizes the use of the iron plate?

(Suppose the thickness of the iron plate is negligible.)

\sqrt[3]{\frac{500}{\pi}}

\sqrt[3]{\frac{1000}{\pi}}

\sqrt[3]{\frac{2000}{\pi}}

\sqrt[3]{\frac{4000}{\pi}}

1 solution

Ilya Prokin
Sep 11, 2017

Surface area (S) is the sum of bottom and top circles ( $\pi r^2$ each) and cyclindric wall ( $2\pi r h$ ). It can be expressed using fixed volume ( $V=\pi r^2 h$ ):

$S = 2\pi r h + 2\pi r^2 = 2 V/r + 2 \pi r^2 \\ S' = -2V/r^2+4\pi r \\ S' = 0 => r = \sqrt[3]{\frac{V}{2\pi}}$

Check if this extrema is max or min:

$S'' = 4V/r^3+4\pi > 0$ when r>0 => $S(\sqrt[3]{\frac{V}{2\pi}})$ is minimum of S.

Lol thats exactly how i did it. There is a question identical to this in the Calculus Done Right course, Linear approximations and Applications

Krishna Karthik - 2 years, 9 months ago

Best Cylinder

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