Hemisphere cut

Calculate the volume of the spherical layer that remains from the hemisphere after the 3 cm section is cut. The height of the hemisphere is 10 cm.

Result

V =  1839.926 cm3

Solution:

v=3 cm r=10 cm  V1=43 π r3=43 3.1416 1034188.7902 cm3 V2=13 π v2 (3 rv)=13 3.1416 32 (3 103)254.469 cm3  V=V1/2V2=4188.7902/2254.4691839.92611839.926 cm3v=3 \ \text{cm} \ \\ r=10 \ \text{cm} \ \\ \ \\ V_{1}=\dfrac{ 4 }{ 3 } \cdot \ \pi \cdot \ r^3=\dfrac{ 4 }{ 3 } \cdot \ 3.1416 \cdot \ 10^3 \doteq 4188.7902 \ \text{cm}^3 \ \\ V_{2}=\dfrac{ 1 }{ 3 } \cdot \ \pi \cdot \ v^2 \cdot \ (3 \cdot \ r - v)=\dfrac{ 1 }{ 3 } \cdot \ 3.1416 \cdot \ 3^2 \cdot \ (3 \cdot \ 10 - 3) \doteq 254.469 \ \text{cm}^3 \ \\ \ \\ V=V_{1}/2 - V_{2}=4188.7902/2 - 254.469 \doteq 1839.9261 \doteq 1839.926 \ \text{cm}^3



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