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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
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B
4
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C
5
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D
6
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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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