Question

A solid is in the shape of a hemisphere surmounted by a cone of the same radius.The diameter of the cone is 18cm and the height of the cone is 14cm.The solid is completely immersed in a cylindrical tub,full of water.If the diameter of the tub is 26 cm and its height is 21cm,find the quantity of water left in the cylindrical tub in litres.​

Answer 1

Answer :-

$\:\\\large{\boxed{\sf{Firstly,\;let's\;understand\;the\;concept\;used\;:-}}}$

Here the concept of Volume of Cylinder, Volume of Cone and Volume of Hemisphere has been used. We see we can calculate the volume of the solid by adding the volume of conical part with the volume of hemispherical part. The we can fund the volume of tub. If we subtract the volume of solid with volume of tub, we can find the quantity of water left.

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$\:\\\large{\boxed{\sf{Volume\;of\;Cone\;=\;\bf{\dfrac{1}{3}\:\times\:\pi r^{2}h}}}}$

$\:\\\large{\boxed{\sf{Volume\;of\;Hemisphere\;=\;\bf{\dfrac{2}{3}\:\times\:\pi r^{3}}}}}$

$\:\\\large{\boxed{\sf{Volume\;of\;Cylinder\;=\;\bf{\pi r^{2}h}}}}$

$\:\\\large{\boxed{\sf{Volume\;of\;Solid\;=\;\bf{Volume\;of\;Cone\;+\;Volume\;of\;Hemisphere}}}}$

$\:\\\large{\boxed{\sf{Quantity\;of\;water\;left\;=\;\bf{Volume\;of\;Tub\;-\;Volume\;of\;Solid}}}}$

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★ Solution :-

» Diameter of cone = Diameter of hemisphere = 18 cm

» Radius of cone = Radius of hemisphere = 9 cm

» Height of cone = 14 m

» Height of Cylinder = 21 cm

» Diameter of Cylinder = 26 cm

» Radius of Cylinder = 13 cm

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~ For the Volume of Cone :-

$\:\\\qquad\large{\sf{:\Longrightarrow\;\;\:Volume\;of\;Cone\;=\;\bf{\dfrac{1}{3}\:\times\:\pi r^{2}h}}}$

$\:\\\qquad\large{\sf{:\Longrightarrow\;\;\:Volume\;of\;Cone\;=\;\bf{\dfrac{1}{3}\:\times\:\dfrac{22}{7}\:\times\:9^{2}\:\times\:14\:\;=\:\;\underline{\underline{1188\;\:cm^{3}}}}}}$

$\:\\\large{\boxed{\boxed{\tt{Volume\;\;of\;\;Cone\;\;=\;\bf{1188\;\;cm^{3}}}}}}$

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~ For the Volume of Hemisphere :-

$\:\\\qquad\large{\sf{:\Longrightarrow\;\;\:Volume\;of\;Hemisphere\;=\;\bf{\dfrac{2}{3}\:\times\:\pi r^{3}}}}$

$\:\\\qquad\large{\sf{:\Longrightarrow\;\;\:Volume\;of\;Hemisphere\;=\;\bf{\dfrac{2}{3}\:\times\:\dfrac{22}{7}\:\times\:(9)^{3}\:\;=\:\;\underline{\underline{1527.42857\;\:cm^{3}}}}}}$

$\:\\\large{\boxed{\boxed{\tt{Volume\;\;of\;\;Hemisphere\;\;=\;\bf{1527.43\;\;cm^{3}}}}}}$

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~ For the Volume of Cylinder :-

$\:\\\qquad\large{\sf{:\Longrightarrow\;\;\:Volume\;of\;Cylinder\;=\;\bf{\pi r^{2}h}}}$

$\:\\\qquad\large{\sf{:\Longrightarrow\;\;\:Volume\;of\;Cylinder\;=\;\bf{\dfrac{22}{7}\:\times\: (13)^{2}\:\times\:21\:\;=\:\;\underline{\underline{11154\:\;cm^{3}}}}}}$

$\:\\\large{\boxed{\boxed{\tt{Volume\;\;of\;\;Tub\;\;=\;\bf{11154\;\;cm^{3}}}}}}$

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~ For the Volume of Solid :-

$\:\\\qquad\large{\sf{:\longmapsto\;\;\:Volume\;of\;Solid\;=\;\bf{Volume\;of\;Cone\;+\;Volume\;of\;Hemisphere}}}$

$\:\\\qquad\large{\sf{:\longmapsto\;\;\:Volume\;of\;Solid\;=\;\bf{1188\;+\;1527.43\;\:=\;\:\underline{\underline{2715.43\;\:cm^{3}}}}}}$

$\:\\\large{\boxed{\boxed{\tt{Volume\;\;of\;\;Solid\;\;=\;\bf{2715.43\;\;cm^{3}}}}}}$

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~ For the Quantity of Water Left in Tub :-

$\:\\\qquad\large{\sf{:\longmapsto\;\;\:Quantity\;of\;water\;left\;=\;\bf{Volume\;of\;Tub\;-\;Volume\;of\;Solid}}}$

$\:\\\qquad\large{\sf{:\longmapsto\;\;\:Quantity\;of\;water\;left\;=\;\bf{11154\;-\;2715.43\;\:=\;\:\underline{\underline{8438.57\;\:cm^{3}}}}}}$

$\:\\\large{\boxed{\boxed{\tt{Quantity\;\;of\;\;Water\;\;left\;\;=\;\bf{8438.57\;\;cm^{3}}}}}}$

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We know that 1 L = 1000 cm³

Then,

=> 8438.57 cm³ = 8.43857 Litres

Hence, water left = 8.43857 Litres.

$\:\\\large{\underline{\underline{\rm{Thus,\;quantity\;of\;water\;left\;in\;tub\;is\;\:\boxed{\bf{8.43857\;\:Litres}}}}}}$

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★ More to know :-

• Volume of Cuboid = Length × Breadth × Height

• Volume of Sphere = (4/3) × πr³

• TSA of Cone = πrL + πr²

• TSA of Cylinder = 2πr² + 2πrh

• TSA of Hemisphere = 3πr²

Answer 2

Solution :

Volume left in the cylindrical vessel = 8.438 litres

Step by step Explanation

Given:

• Radius of a cone, r = 18/2= 9cm
• Height of cone, h = 14cm
• Radius of hemisphere ,r = 9cm

Radius of base of the cone=Radius of the hemisphere. Because the solid is in the form of a hemisphere surmounted by a cone

To Find :

The quantity of water left in the cylindrical tub in litres.If the solid is completely immersed in a cylindrical tub,full of water.If the diameter of the tub is 26 cm and its height is 21cm.

Solution:

Volume of the solid

= Volume of hemisphere + volume of cone

$\sf=\dfrac{1}{3}\pi\:r^2h+\dfrac{2}{3}\pi\:r^3$

$\sf=\dfrac{1}{3}\pi\:r^2(h+2r)$

Now put the given values

$\sf=\dfrac{1}{3}\times\dfrac{22}{7}(9)(14+18)$

$\sf=\dfrac{22\times9\times32}{7\times3}$

$\sf=\dfrac{57024}{21}$

$\sf=2715.42cm^3$

Now,

Radius of the base of the cylindrical vessel, r=13cm

Height of the cylindrical vessel,h = 21cm

Volume of the water in the cylindrical vessel

$\sf=\pi\:r^2h$

Put the given values

$\sf=\dfrac{22}{7}\:(13)^2(21)$

$\sf=\dfrac{22\times169\times21}{7}$

$\sf=\dfrac{78078}{7}$

$\sf=11,154cm^3$

According to the question :

The solid is completely submerged in the cylindrical vessel full of water, then

Volume of the water left in the vessel

=Volume of the water in the cylindrical vessel - volume of solid

=(11,154-2715.42)

=8438.58 cm³

= 8.438 litres

Therefore , the quantity of water left in the cylindrical tub in litres is 8.438 l