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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.


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