Question

(4)The radius and the height of a cylinder are in the ratio 7: 3. If the volume of the cylinder
is 12474 cm, Find the curved surface area and total surface area of the cylinder.​

Answer 1

Answer :-

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Question\;:-}}}$

Here the concept of Volume, TSA and CSA of Cylinder has been used. Here we see that the height and radius are given in ratio. So first we will find the constant by which both height and radius should be multiplied to get their original value since we are given volume. Then we can find slant height and then TSA and CSA of cylinder.

Let's do it !!

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★ Formula Used :-

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

$\\\;\boxed{\sf{CSA\;of\;Cylinder\;=\;\bf{2\pi rh}}}$

$\;\boxed{\sf{TSA\;of\;Cylinder\;=\;\bf{2\pi rh \;+\;2\pi r^{2}}}}$

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★ Solution :-

Given,

» Ratio of radius and height = h : r = 7 : 3

» Volume of Cylinder = 12474 cm³

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~ For the height and radius of cylinder ::

• h : r = 7 : 3

Let the constant by which both height and radius shall be multiplied be x .

Then,

• Radius of Cylinder = 7x

• Height of Cylinder = 3x

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~ For the value of x ::

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

$\\\;\;\;\;\sf{:\rightarrow\;\;\dfrac{22}{7}\:\times\: (7x)^{2}\:\times\:3x\;=\;\bf{Volume\;of\;Cylinder}}$

$\\\;\;\;\;\sf{:\rightarrow\;\;\dfrac{22}{7}\:\times\: 49x^{2}\:\times\:3x\;=\;\bf{12474}}$

$\\\;\;\;\;\sf{:\rightarrow\;\;\dfrac{22}{7}\:\times\: 49\:\times\:3x^{3}\;=\;\bf{12474}}$

$\\\;\;\;\;\sf{:\rightarrow\;\;x^{3}\;=\;\bf{\dfrac{12474\;\times\;7}{22\;\times\;49\;\times\;3}}}$

$\\\;\;\;\;\sf{:\rightarrow\;\;x^{3}\;=\;\bf{\dfrac{87318}{3234}}}$

$\\\;\;\;\;\sf{:\rightarrow\;\;x^{3}\;=\;\bf{27}}$

$\\\;\;\;\;\sf{:\rightarrow\;\;x\;=\;\bf{\sqrt{27}}}$

$\\\;\;\;\;\sf{:\rightarrow\;\;x\;=\;\bf{3}}$

Hence, x = 3

Now by applying values of x in height and radius ratio, we get,

• Radius of cylinder = r = 7x = 7(3) = 21 cm

$\\\;\underline{\boxed{\tt{Radius\;\;of\;\;Cylinder\;=\;\bf{21\;\;cm}}}}$

• Height of cylinder = h = 3x = 3(3) = 9 cm

$\\\;\underline{\boxed{\tt{Height\;\;of\;\;Cylinder\;=\;\bf{9\;\;cm}}}}$

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~ For the CSA of Cylinder ::

$\\\;\;\;\;\sf{:\Longrightarrow\;\;CSA\;of\;Cylinder\;=\;\bf{2\pi rh}}$

$\\\;\;\;\;\sf{:\Longrightarrow\;\;CSA\;of\;Cylinder\;=\;\bf{2\;\times\;\dfrac{22}{7}\;\times\;21\;\times\;9}}$

$\\\;\;\;\;\sf{:\Longrightarrow\;\;CSA\;of\;Cylinder\;=\;\bf{2\;\times\;22\;\times\;3\;\times\;9}}$

$\\\;\;\;\;\sf{:\Longrightarrow\;\;CSA\;of\;Cylinder\;=\;\bf{1188\;\;cm^{2}}}$

$\\\;\large{\underline{\underline{\rm{CSA\;of\;Cylinder\;is\;\;\boxed{\bf{1188\;\;cm^{2}}}}}}}$

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~ For the TSA of Cylinder ::

$\\\;\;\;\;\sf{:\mapsto\;\;TSA\;of\;Cylinder\;=\;\bf{(2\;\times\;\dfrac{22}{7}\;\times\;21\;\;\times\;9)\;+\;(2\;\times\;\dfrac{22}{7}\;\times\;(21)^{2})}}$

$\\\;\;\;\;\sf{:\mapsto\;\;TSA\;of\;Cylinder\;=\;\bf{(1188)\;+\;(2772)}}$

$\\\;\;\;\;\sf{:\mapsto\;\;TSA\;of\;Cylinder\;=\;\bf{3960\;\;cm^{2}}}$

$\\\;\large{\underline{\underline{\rm{TSA\;of\;Cylinder\;is\;\;\boxed{\bf{3960\;\;cm^{2}}}}}}}$

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★ More to know :-

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

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

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

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

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

Answer 2

Answer:

Given :-

The radius and the height of a cylinder are in the ratio 7: 3. If the volume of the cylinder

is 12474 cm, Find the curved surface area and total surface area of the cylinder.​

To find :-

$\color{orange} {\Rightarrow}$curved surface area and total surface area of the cylinder.​

Solution :-

$\color{orange} {\Rightarrow}$let the radius of cylinder= r cm

$\color{orange} {\Rightarrow}$the height = h cm

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$\color{orange} {\Rightarrow}$ratio of r and h = 7:3

$\color{orange} {\Rightarrow}$r / h = 7 /3 => r = 7h/ 3

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$\color{orange} {\Rightarrow}$volume = 12474

$\color{orange} {\Rightarrow}$πr^2h = 12474

$\color{orange} {\Rightarrow}$(22/ 7) × ( 7h/ 3)^2 × h = 12474

$\color{orange} {\Rightarrow}$154 h^3 / 9 = 12474

$\color{orange} {\Rightarrow}$h^3 =( 12474 × 9)/ 154

$\color{orange} {\Rightarrow}$h^3 = 81 × 9 = 729

$\color{orange} {\Rightarrow}$h = 9 cm , r = 7h/ 3 = (7 × 9)/ 3 = 21 cm

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$\color{orange} {\Rightarrow}$c.s.a of cylinder = 2πrh

$\color{orange} {\Rightarrow}$= (2 × 22 × 21 × 9 ) ÷ 7 = 1188 cm^2

$\color{orange} {\Rightarrow}$total surface area of cylinder

$\color{orange} {\Rightarrow}$= 2πr ( r + h ) = 2 × (22/ 7) × 21 ( 21 + 9)

$\color{orange} {\Rightarrow}$= 44 × 3 × 30 = 3960 cm^2

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Answer:-

$\color{orange} {\Rightarrow}$r=21 cm

$\color{orange} {\Rightarrow}$h=9 cm

$\color{orange} {\Rightarrow}$c.s.a = 1188cm^2

$\color{orange} {\Rightarrow}$t.s.a = 3960 cm^2

Step-by-step explanation:

Some more things to know :-

Shapes and it volume.

$\color{red}\looparrowright$Rectangular Solid or Cuboid = V = l × w × h

$\color{red}\looparrowright$Cube= V = a^3

$\color{red}\looparrowright$Cylinder =V = πr2h

$\color{red}\looparrowright$Prism = V = B × h

$\color{red}\looparrowright$Sphere =V = (4⁄3)πr3

$\color{red}\looparrowright$Pyramid =V = (1⁄3) × B × h

$\color{red}\looparrowright$Right Circular Cone= V = (1⁄3)πr2h

$\color{red}\looparrowright$Ellipsoid= V = (4⁄3) × π × a × b × c

$\color{red}\looparrowright$Tetrahedron= V = a3⁄ (6 √2)

$\color{red}\looparrowright$Square or Rectangular Pyramid= V = (1⁄3) × l × w × h

i hope it helps uh.....