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.

Correct result:

r =  9.9983 cm
v =  9.9983 cm

Solution:

$$\begin{gathered} V=3140{\text{cm}}^3 \\ V=\pi r^{2}v \\ v=V\mathrm{/}(\pi r^2) \\ S=\pi r^2+2\pi rv \\ S=\pi r^2+2\pi rV\mathrm{/}(\pi r^2) \\ S=\pi r^2+2\cdot 3140\mathrm{/}r \\ {S}^{\mathrm{\prime }}=2\pi r-6280\mathrm{/}{r}^2 \\ {S}^{\mathrm{\prime }}=0 \\ 2\pi r=6280\mathrm{/}{r}^2 \\ 2\pi r^3=6280 \\ r=\sqrt[3]{6280\mathrm{/}(2\pi )}=\sqrt[3]{6280\mathrm{/}(2\cdot 3.1416)}=9.9983\text{ cm} \end{gathered}$$

$v=V\mathrm{/}(\pi \cdot r^2)=3140\mathrm{/}(3.1416\cdot 9.9983^2)=9.9983\text{ cm}$

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