# Cylindrical container

An open-topped cylindrical container has a volume of V = 3140 cm3. Find the cylinder dimensions (radius of base r, height v) so that the least material is needed to form the container.

Result

r =  9.998 cm
v =  9.999 cm

#### Solution:

$V=3140 \ \text{cm}^3 \ \\ V=\pi r^2 \ v \ \\ v=V/(\pi r^2) \ \\ S=\pi r^2 + 2 \ \pi r v \ \\ S=\pi r^2 + 2 \ \pi r V /(\pi r^2) \ \\ S=\pi r^2 + 2 \cdot \ 3140 / r \ \\ S'=2 \ \pi r - 6280/r^2 \ \\ S'=0 \ \\ 2 \ \pi r=6280/r^2 \ \\ 2 \ \pi r^3=6280 \ \\ r=\sqrt[3]{ 6280/(2 \pi)}=\sqrt[3]{ 6280/(2 \cdot \ 3.1416)} \doteq 9.9983 \ \text{cm} \doteq 9.998 \ \text{cm}$
$v=V/(\pi \cdot \ r^2)=3140/(3.1416 \cdot \ 9.9983^2) \doteq 9.9989 \doteq 9.999 \ \text{cm}$

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