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A hollow cylinder has solid hemisphere inward at one end and on the other end it is closed with a flat circular plate. The height of water is 10 cm when flat circular surface is downward. Find the level of water, when it is inverted upside down, common diameter is 7 cm and height of cylinder is 20 cm.

Mensuration

ICSE

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Answer

Given,

For Cylinder :

Height (H) = 20 cm, Radius (R) = 3.5 cm

For Hemisphere :

Radius (r) = Radius of cylinder (R) = 3.5 cm

A hollow cylinder has solid hemisphere inward at one end and on the other end it is closed with a flat circular plate. The height of water is 10 cm when flat circular surface is downward. Find the level of water, when it is inverted upside down, common diameter is 7 cm and height of cylinder is 20 cm. Cylinder, Cone, Sphere, Concise Mathematics Solutions ICSE Class 10.

When flat surface is downward, then height of water (h) = 10 cm.

So, when circular surface part will be downward then,

Let height upto which water fills be h1.

Volume of cylinder (upto height h1) = Volume of cylinder (upto height h) + Volume of hemisphere

⇒πR2h1=πR2h+23πr3⇒πR2h1=πR2h+23πR3[As,r=R]⇒πR2h1=πR2(h+23R)⇒h1=h+23R⇒h1=10+23×3.5⇒h1=10+73⇒h1=30+73⇒h1=373=1213=12.33 cm.\Rightarrow πR^2h1 = πR^2h + \dfrac{2}{3}πr^3 \\[1em] \Rightarrow πR^2h1 = πR^2h + \dfrac{2}{3}πR^3 [As, r = R] \\[1em] \Rightarrow πR^2h1 = πR^2\Big(h + \dfrac{2}{3}R\Big) \\[1em] \Rightarrow h1 = h + \dfrac{2}{3}R \\[1em] \Rightarrow h1 = 10 + \dfrac{2}{3}\times 3.5 \\[1em] \Rightarrow h1 = 10 + \dfrac{7}{3} \\[1em] \Rightarrow h1 = \dfrac{30 + 7}{3} \\[1em] \Rightarrow h1 = \dfrac{37}{3} = 12\dfrac{1}{3} = 12.33\text{ cm}.

Hence, the level of water, when cylinder is inverted upside down = 12.33 cm.

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