  Question

A sphere of diameter 18 cm is dropped into a cylindrical vessel of diameter 36cm, partly filled with water. If the sphere is completely submerged, then the water level rises (in cm) by

A
3  B
4  C
5  D
6  Solution

The correct option is A $$3$$Let $$r$$ & $$R$$ be the radius of the sphere and cylindrical vessel respectively.Now,  $$2r=18cm\Rightarrow r=9cm$$         $$2R=36cm\Rightarrow R=18cm$$Let the rise in water level in the cylindrical vessel be $$'h'$$ cm.Volume of sphere $$=\dfrac { 4 }{ 3 } \pi { r }^{ 3 }$$Volume of liquid displaced in the cylindrical vessel $$=\pi { R }^{ 2 }h$$If the sphere is completely submerged in the vessel, then volume of liquid displace in the cylindrical vessel $$=$$ Volume of the sphere$$\therefore \quad \pi { R }^{ 2 }h=\dfrac { 4 }{ 3 } \pi { r }^{ 3 }\Rightarrow { \left( 18 \right) }^{ 2 }\times h=\dfrac { 4 }{ 3 } \times { \left( 9 \right) }^{ 3 }$$$$\Rightarrow \quad h=\dfrac { 4\times { \left( 9 \right) }^{ 3 } }{ 3\times { \left( 18 \right) }^{ 2 } } =3cm$$Thus, the rise in water level in the cylindrical vessel is $$3cm.$$Mathematics

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