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Question

A hemispherical bowl of internal diameter $$36\ cm$$ contains a liquid. This liquid is to be filled in cylindrical bottles of radius $$3\ cm$$ and height $$6\ cm$$. How many bottles are required to empty the bowl?


Solution

We have
Radius of hemispherical bowl$$=\dfrac{36}{2}=18cm$$

Volume of hemispherical bowl$$=\cfrac { 2 }{ 3 } \pi \times { (18) }^{ 3 }{ cm }^{ 3 }\left[ V=\cfrac { 2 }{ 3 } \pi { r }^{ 3 }\quad  \right]  $$

Radius of a cylindrical bottle $$=3cm$$
Height of a cylindrical bottle $$=6cm$$

Volume of a cylindrical bottle $$=\left( \pi \times { 3 }^{ 2 }\times 6 \right) { cm }^{ 3 }$$   ....... $$v=\pi r^2 h$$
Suppose $$x$$ bottles are required to empty the bowl
Volume of $$x$$ cylindrical bottles $$=\left( \pi \times 9\times 6\times x \right) { cm }^{ 3 }$$

Clearly volume of liquid in $$x$$ bottles = Volume of bowl
$$\Rightarrow \pi \times 9\times 6\times x=\cfrac { 2 }{ 3 } \pi \times { (18) }^{ 3 }$$
$$\Rightarrow x=72\quad $$

$$\therefore$$ Bottles required to empty the bowl$$=72$$

Mathematics

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