Question

2. A cylindrical vessel has diameter equal to its height. It is full of milk. This milk is poured into two cylindrical
vessels of same size of height 21 cm and radius 21 cm. If these vessels are filled completely find the radius
of the first cylindrical vessel.​

Answer 1

Given,

A cylindrical vessel has diameter equal to its height. It is full of milk. This milk is poured into two cylindrical  vessels of same size of height 21 cm and radius 21 cm.

To find :

Find the radius  of the first cylindrical vessel.​

Solution :

Let the diameter of first vessel be d cm

∴ Radius of first vessel = d/2 cm

∴ Height of first vessel = d cm

So volume of first vessel :

⇒ Volume of cylinder = πr²h

⇒ Volume of 1st vessel = π(d/2)² * d

⇒ Volume of 1st vessel = π(d²/4) * d

⇒ Volume of 1st vessel = πd³/4 cm³

Now radius of two same vessels = 21 cm each

Height of 2 vessels = 21 cm each.

∴ Volume of 2 vessels = 2 * [π(21)² * 21]

⇒ Volume of 2 vessels = 2 * [π * 9261]

⇒ Volume of 2 vessels = 18522π cm³

Now atq,

⇒ πd³/4 = 18522π

⇒ d³/4 = 18522

⇒ d³ = 4 * 18522

⇒ d³ = 74088

⇒ d = ∛74088

⇒ d = 42 cm

∴ Diameter of 1st vessel = 42 cm

∴ Radius of 1st vessel = 42/2 cm

∴ Radius of 1st vessel = 21 cm

Therefore,

Radius of the first cylindrical vessel = 21 cm.

Answer 2

Answer :-

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

Here the concept of Volume of Cylinder has been used. We see that milk in the larger cylindrical vessel has been poured into two smaller cylinders of equal sizes. Now, volume of both smaller cylinders will be equal to the volume of single large cylinder. Since, two smaller cylinders are equal, their volume will be twice the volume of each smaller cylindrical vessel.

Let's do it !!

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★ Formula Used :-

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

$\:\\\large{\boxed{\sf{2\:\times\:Volume\;of\;Smaller\;Vessel\;\;=\;\;\bf{Volume\;of\;Bigger\;Vessel}}}}$

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★ Question :-

A cylindrical vessel has diameter equal to its height. It is full of milk. This milk is poured into two cylindrical vessels of same size of height 21 cm and radius 21 cm. If these vessels are filled completely find the radius of the first cylindrical vessel.

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★ Solution :-

Given,

» Diameter of larger vessel = Height of larger vessel

» Radius of Larger Vessel = ½ × Diameter

» Radius of Smaller Vessel = 21 cm

» Height of Smaller Vessel = 21 cm

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~ For the Volume of Each Smaller Vessel :-

$\:\\\qquad\large{\sf{:\longrightarrow\;\:\;Volume\;of\;Cylinder_{(smaller\;vessel)}\;\:=\;\bf{\pi r^{2}h}}}$

$\:\\\qquad\large{\sf{:\longrightarrow\;\:\;Volume\;of\;Cylinder_{(smaller\;vessel)}\;\:=\;\bf{\dfrac{22}{7}\:\times\:(21)^{2}\:\times\:21\;\:=\;\:\underline{\underline{29106\;\;cm^{3}}}}}}$

$\:\\\large{\boxed{\boxed{\tt{Volume\;\;of\;\;Smaller\;\;Vessel\;\;=\;\bf{29106\;\;cm^{3}}}}}}$

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~ For the Volume of Bigger Vessel:-

• Let the Radius of the Bigger Vessel be r.

Then, Height of bigger vessel will be 2r.

$\:\\\qquad\large{\sf{:\longrightarrow\;\;\:Volume\;of\;Cylinder_{(Bigger\;vessel)}\;\:=\;\bf{\pi r^{2}h}}}$

$\:\\\qquad\large{\sf{:\longrightarrow\;\;\:Volume\;of\;Cylinder_{(Bigger\:vessel)}\;\:=\;\bf{\dfrac{22}{7}\:\times\:r^{2}\:\times\:2r\:\;=\:\;\underline{\underline{\dfrac{44}{7}\:r^{3}\;\;cm^{3}}}}}}$

$\:\\\large{\boxed{\boxed{\tt{Volume\;\;of\;\;Bigger\;\;Vessel\;\;=\;\bf{\dfrac{44}{7}\:r^{3}\;\;cm^{3}}}}}}$

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~ For the Radius of the Bigger Vessel :-

$\:\\\qquad\large{\sf{:\Longrightarrow\;\;\:2\:\times\:Volume\;of\;Smaller\;Vessel\;\;=\;\;\bf{Volume\;of\;Bigger\;Vessel}}}$

$\:\\\qquad\large{\sf{:\Longrightarrow\;\;\:2\:\times\:29106\;\;=\;\;\bf{\dfrac{44}{7}\:\times\:r^{3}}}}$

$\:\\\qquad\large{\sf{:\Longrightarrow\;\;\:r^{3}\;\;=\;\;\bf{\dfrac{2\:\times\:29106\:\times\:7}{44}\:\;=\;\: 9261}}}$

$\:\\\qquad\large{\sf{:\Longrightarrow\;\;\:r^{3}\;\;=\;\;\bf{\sqrt[3]{9261}\:\;=\;\:\underline{\underline{21\;\;cm}}}}}$

$\:\\\large{\boxed{\boxed{\tt{Radius\;\;of\;\;Larger\;\;Vessel\;\;=\;\bf{21\;\;cm}}}}}$

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~ For the Height and Radius of Bigger Vessel :-

$\:\\\sf{\rightarrow\;\;Radius\;of\;Larger\;Vessel\;=\;\bf{r}\;=\;\bf{21\;\;cm}}$

$\:\\\sf{\rightarrow\;\;Radius\;of\;Larger\;Vessel\;=\;\bf{2r}\;=\;2\:\times\:21\;\;cm\;\;=\;\:\bf{42\;\;cm}}$

$\:\\\large{\underline{\underline{\rm{Thus,\;radius\;of\;larger\;vessel\;is\;\;\boxed{\bf{21\;\;cm}}}}}}$

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★ More to know :-

• CSA of Cylinder = 2πrh

• TSA of Cylinder = 2πrh + 2πr²

• CSA of Cone = πrl

• TSA of Cone = πrl + πr²

• Volume of Cone = ⅓ × πr²h

• Volume of Cube = (Side)³

• Volume of Cuboid = Length × Breadth × Height

• Volume of Hemisphere = ⅔ × πr³