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## Cylinder problem (1)

Find cylinder with lowest surface area which volume is V .

# Solution :

The volume of the cylinder is $V = \pi r^2 h$ (r-radius, h-height) and height of cylinder will be $h = \frac{V}{ \pi r^2 }$

The surface area of cylinder is $S = 2 \pi r^2 + 2 \pi r h = 2 \pi r^2 + 2V/r$

We can write it as function    $S(r) = 2 \pi r^2 + 2V/r$

Derivative of function is $S'(r) = 4 \pi r - 2V/r^2$

If $S'(r)=0$ then $r= \sqrt[3]{ \frac{V}{2 \pi } }$

$h = \frac{V}{ \pi r^2 } = 2r$

if h=2r the cylinder will have lowest surface area.

Categories: Cylinder