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 answer:

r =  9.9983 cm
v =  9.9983 cm

Step-by-step explanation:

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

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

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