Question

2. A cylindrical vessel has diameter equal to
s 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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Answer 1

Answer :-

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

Here the concept of Volume of Cuboid had been used. First we can find the Volume of Box and then we can find the volume of each soap. After that we cab divide the volume of box by volume of each soap to find the number of soaps required, since volume can neither be created nor be destroyed.

Let's do it !!

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★ Formula Used :-

$\:\\\large{\boxed{\sf{Volume\;of\;Cuboid\;=\;\bf{Length\:\times\:Breadth\:\times\:Height}}}}$

$\:\\\large{\boxed{\sf{Number\;of\;soap\;cakes\;=\;\bf{\dfrac{Volume\;of\;box}{Volume\;of\;each\;soap\;cake}}}}}$

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★ Question :-

Find the number of cakes of soap each measuring 9 cm x 5 cm x 2 cm that can be placed in a big cuboidal box measuring 1.5 m x 0.9 m x 0.5 m.

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★ Solution :-

Given,

» Dimensions of box = 1.5 m × 0.9 m × 0.5 m

= 150 m × 90 m × 50 m

» Dimensions of each soap cake

= 9 cm × 5 cm × 2 cm

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~ For the Volume of Box :-

$\:\\\qquad\large{\sf{:\longrightarrow\;\;\:Volume\;of\;Cuboid_{(Box)}\;=\;\bf{Length\:\times\:Breadth\:\times\:Height}}}$

$\:\\\qquad\large{\sf{:\longrightarrow\;\;\:Volume\;of\;Cuboid_{(Box)}\;=\;\bf{150\:\times\:90\:\times\:50\;\:=\;\:\underline{\underline{675000\;\;cm^{3}}}}}}$

$\:\\\large{\boxed{\boxed{\tt{Volume\;of\;box\;=\;\bf{675000\;\;cm^{3}}}}}}$

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~ For the Volume of Each Soap Cake :-

$\:\\\qquad\large{\sf{:\longrightarrow\;\;\:Volume\;of\;Cuboid_{(Each\:soap\:cake)}\;=\;\bf{Length\:\times\:Breadth\:\times\:Height}}}$

$\:\\\qquad\large{\sf{:\longrightarrow\;\;\:Volume\;of\;Cuboid_{(Each\:soap\:cake)}\;=\;\bf{9\:\times\:5\:\times\:2\;\:=\;\:\underline{\underline{90\;\;cm^{3}}}}}}$

$\:\\\large{\boxed{\boxed{\tt{Volume\;of\;Each\;soap\;cake\;=\;\bf{90\;\;cm^{3}}}}}}$

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~ For the Number of Each Soap Cakes :-

$\:\\\qquad\large{\sf{:\Longrightarrow\;\;\:Number\;of\;soap\;cakes\;=\;\bf{\dfrac{Volume\;of\;box}{Volume\;of\;each\;soap\;cake}}}}$

$\:\\\qquad\large{\sf{:\Longrightarrow\;\;\:Number\;of\;soap\;cakes\;=\;\bf{\dfrac{675000}{90}\;\:=\;\:\underline{\underline{7500}}}}}$

$\:\\\large{\underline{\underline{\rm{Thus,\;number\;of\;soap\;cakes\;required\;are\;\:\boxed{\bf{7500}}}}}}$

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★ More to know :-

• Volume of Cube = (Side)³

• Volume of Cone = ⅓ × πr²h

• Volume of Cylinder = πr²h

• Volume of Sphere = ⅔ × πr³

• Volume of Hemisphere = (4/3) × πr³

Answer 2

Answer:

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \large \sf \bold{Given:-}$

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.

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$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \large\sf{To \: find :-}$

Find the radius  of the first cylindrical vessel.

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$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \large \sf{Solution :-}$

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

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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 =

18522πcm³

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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.

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