Cone container

Rotary cone-shaped container has a volume 1000 cubic cm and a height 12 cm. Calculate how much metal we need for making this package.

Result

S =  669.043 cm2

Solution:

V=1000 cm3 h=12 cm  V=13πr2 h r=3⋅ V/h/π=3⋅ 1000/12/3.1416≐8.9206 cm  s=r2+h2=8.92062+122≐14.9525 cm  S1=π⋅ r2=3.1416⋅ 8.92062=250 cm2 S2=π⋅ r⋅ s=3.1416⋅ 8.9206⋅ 14.9525≐419.0434 cm2 S=S1+S2=250+419.0434≐669.0434≐669.043 cm2V=1000 \ \text{cm}^3 \ \\ h=12 \ \text{cm} \ \\ \ \\ V=\dfrac{ 1 }{ 3 } \pi r^2 \ h \ \\ r=\sqrt{ 3 \cdot \ V/h/\pi }=\sqrt{ 3 \cdot \ 1000/12/3.1416 } \doteq 8.9206 \ \text{cm} \ \\ \ \\ s=\sqrt{ r^2+h^2 }=\sqrt{ 8.9206^2+12^2 } \doteq 14.9525 \ \text{cm} \ \\ \ \\ S_{1}=\pi \cdot \ r^2=3.1416 \cdot \ 8.9206^2=250 \ \text{cm}^2 \ \\ S_{2}=\pi \cdot \ r \cdot \ s=3.1416 \cdot \ 8.9206 \cdot \ 14.9525 \doteq 419.0434 \ \text{cm}^2 \ \\ S=S_{1}+S_{2}=250+419.0434 \doteq 669.0434 \doteq 669.043 \ \text{cm}^2

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