Question

14. If the radius of a cylinder is r and the height is h,
how will the volume change, if the (a) height is
doubled (b) height is doubled and the radius is
halved and (c) height is same and the radius is
halved..​

Answer 1

Answer :-

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Concept\;of\;Solving\;::}}}$

Here the concept of Volume of Cylinder has been used. We are given certain conditions of change in dimensions of cylinder. We can apply them in the formula of Volume of Cylinder and find our answer.

Let's do it !!

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★ Formula used :-

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

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★ Solution :-

Given,

» Radius of the cylinder = r

» Height of the cylinder = h

Then, without changing dimensions of Cylinder, its volume will be :-

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

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a.) ~ For the Volume change of cylinder when height is doubled ::

• Height of the cylinder = 2h

$\\\;\;\;\;\:\sf{:\Longrightarrow\;\;\;New\;Volume\;of\;Cylinder\;=\;\bf{\pi r^{2}\:\times\:Height}}$

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

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

$\\\;\;\;\;\:\underline{\boxed{\sf{:\mapsto\;\;\;New\;Volume\;of\;Cylinder\;=\;\bf{2(Volume\;of\;Original\;Cylinder)}}}}$

$\\\;\large{\underline{\underline{\rm{Thus,\;when\;the\;height\;of\;cylinder\;is\;doubled\;then\;its\;volume\;becomes\;\boxed{\bf{Twice}}}}}}$

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b.) ~ For the Volume change of cylinder when height is doubled and radius is halved ::

• Height of Cylinder = 2h

• Radius of Cylinder = ½ × r

$\\\;\;\;\;\:\sf{:\Longrightarrow\;\;\;New\;Volume\;of\;Cylinder\;=\;\bf{\pi\:\times\:(radius)^{2}\:\times\:Height}}$

$\\\;\;\;\;\:\sf{:\Longrightarrow\;\;\;Volume\;of\;Cylinder\;=\;\bf{\pi\:\times\:(\dfrac{r}{2})^{2}\:\times\:2h}}$

$\\\;\;\;\;\:\sf{:\Longrightarrow\;\;\;Volume\;of\;Cylinder\;=\;\bf{\pi\:\times\:\dfrac{r^{2}}{4}\:\times\:2h}}$

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

$\\\;\;\;\;\:\sf{:\Longrightarrow\;\;\;Volume\;of\;Cylinder\;=\;\bf{\dfrac{1}{2}\:(\pi r^{2}h)}}$

$\\\;\;\;\;\:\underline{\boxed{\sf{:\mapsto\;\;\;New\;Volume\;of\;Cylinder\;=\;\bf{\dfrac{1}{2}(Volume\;of\;Original\;Cylinder)}}}}$

$\;\large{\underline{\underline{\rm{Thus,\;when\;the\;height\;is\;twice\;and\;radius\;is\;halved\;then\;volume\;is\;\boxed{\bf{Halved}}}}}}$

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c.) ~ For Volume of cylinder when Radius is halved ::

• Radius of the cylinder = ½ × r

$\\\;\;\;\;\:\sf{:\Longrightarrow\;\;\;New\;Volume\;of\;Cylinder\;=\;\bf{\pi\:\times\:(radius)^{2}\:\times\:h}}$

$\\\;\;\;\;\:\sf{:\Longrightarrow\;\;\;Volume\;of\;Cylinder\;=\;\bf{\pi\:\times\:(\dfrac{r}{2})^{2}\:\times\:h}}$

$\\\;\;\;\;\:\sf{:\Longrightarrow\;\;\;Volume\;of\;Cylinder\;=\;\bf{\pi\:\times\:\dfrac{r^{2}}{4}\:\times\:h}}$

$\\\;\;\;\;\:\sf{:\Longrightarrow\;\;\;Volume\;of\;Cylinder\;=\;\bf{\dfrac{1}{4}\:(\pi r^{2}h)}}$

$\\\;\;\;\;\:\underline{\boxed{\sf{:\mapsto\;\;\;New\;Volume\;of\;Cylinder\;=\;\bf{\dfrac{1}{4}(Volume\;of\;Original\;Cylinder)}}}}$

$\\\;\large{\underline{\underline{\rm{Thus,\;when\;the\;radius\;of\;the\;cylinder\;is\;halved\;then\;its\;volume\;is\;\boxed{\bf{Reduced\;by\;\dfrac{1}{4}\;times}}}}}}$

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★ More formulas to know :-

$\\\;\sf{\leadsto\;\;\;Volume\;of\;Cube\;=\;(Side)^{3}}$

$\\\;\sf{\leadsto\;\;\;Volume\;of\;Cuboid\;=\;Length\:\times\:Breadth\:\times\:Height}$

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

$\\\;\sf{\leadsto\;\;\;Volume\;of\;Hemisphere\;=\;\dfrac{2}{3}\:\pi r^{3}}$

$\\\;\sf{\leadsto\;\;\;Volume\;of\;Sphere\;=\;\dfrac{4}{3}\:\pi r^{3}}$

Answer 2

➠Given:—

• Radius of the cylinder is r.
• Height of the cylinder is h,

➠To find out:—

How many times will the volume change if

• (a) height is doubled.
• (b) height is doubled and the radius is halve.
• (c) height is same and the radius is halve.

➠Solution:—

➠Case —1 :-When the height is doubled

New height of cylinder = 2h

Radius of cylinder =r

⇒Volumeᶜʸˡᶤᶰᵈᵉʳ=πr²(2h) cubic.unit

⇒Volumeᶜʸˡᶤᶰᵈᵉʳ=2(πr²h) cubic.unit

Thus the volume of cylinder doubled when its height is doubled.

➠Case —2 :- When height is doubled and radius is halved.

New height of cylinder =2h

New radius of cylinder =(r/2)

⇒Volumeᶜʸˡᶤᶰᵈᵉʳ=π (r/2)²× 2h cubic.unit

⇒Volumeᶜʸˡᶤᶰᵈᵉʳ= π ×r²/4×2h cubic.unit

⇒Volumeᶜʸˡᶤᶰᵈᵉʳ= 1/2 πr²h cubic.unit

Thus the volume of cylinder is halve when its height is doubled and its radius is halved.

➠Case–3 :-When height is same and the radius is halved.

Height of cylinder =h

New radius of cylinder= (r/2)

⇒Volumeᶜʸˡᶤᶰᵈᵉʳ=π(r/2)²h cubic.unit

⇒Volumeᶜʸˡᶤᶰᵈᵉʳ= π×r²/4×h cubic.unit

⇒Volumeᶜʸˡᶤᶰᵈᵉʳ=1/4 π r²h cubic.unit

Thus the volume of cylinder is 1/4th when the height is same and radius is halved.