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Question

If a cone of radius $$10cm$$ is divided into two parts by drawing a plane through the mid-point of its axis, parallel to its base. Compare the volumes of the two parts.


Solution

Volume of cone$$=\cfrac { 1 }{ 3 } \pi { r }_{ 1 }^{ 2 }{ h }_{ 1 }=\cfrac { 1 }{ 3 } \pi { \left( \cfrac { r }{ 2 }  \right)  }^{ 2 }\times \cfrac { h }{ 2 } $$
$$=\cfrac { 1 }{ 3 } \pi { \left( \cfrac { 10 }{ 2 }  \right)  }^{ 2 }\times \cfrac { h }{ 2 } $$
$$=\cfrac { 25\pi h }{ 6 } cm^3$$
Volume of frustum ABCD$$=\cfrac { 1 }{ 3 } \pi { h }_{ 2 }\left( { R }^{ 2 }+{ r }^{ 2 }+Rr \right) $$
$$=\cfrac { 1 }{ 3 } \pi \times \cfrac { h }{ 2 } \left( { 10 }^{ 2 }+{ 5 }^{ 2 }+10\times 5 \right) $$
$$=\cfrac { 175\pi h }{ 6 } $$
Required ratio$$=\cfrac { \cfrac { 25\pi h }{ 6 }  }{ \cfrac { 175\pi h }{ 6 }  } =\cfrac { 25 }{ 175 } =\cfrac { 1 }{ 7 } =1:7$$

963906_976179_ans_80ca1527da9a4ae69aa242c92d2fdec4.png

Mathematics

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