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Volume of Hollow Cylinder

A hollow cyli...

Question

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 the cylinder is 20 cm.

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Solution

Cylinder: $Height = 20 \ cm, Radius = 3.5 \ cm$

Sphere: $Radius = 3.5 \ cm$

Volume of water $= \pi r^2 h =$ $\frac{22}{7}$ $\times$ $(\frac{7}{2})$ $^2 \times 10 = 385 \ cm^3$

Let the height of water $= h$ when cylinder is upside down

$\Rightarrow \pi r^2 h = 385 +$ $\frac{1}{2}$ $\times$ $\frac{4}{3}$ $\pi \times 3.5^3$

$\Rightarrow$ $\frac{22}{7}$ $\times 3.5^2 \times h = 385 +$ $\frac{1}{2}$ $\times$ $\frac{4}{3}$ $\pi \times 3.5^3$

$\Rightarrow h =$ $\frac{385 \times 7}{22 \times 3.5^2}$ $+$ $\frac{22 \times 3.5^3 \times 7}{3 \times 22 \times 3.5^2}$

$\Rightarrow h = 10 +$ $\frac{7}{3}$ $=$ $12$ $\frac{1}{3}$ $\ cm$

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