Question

A hemispherical bowl of radius 18cm is full of fruit juice. The juice is to be filled in cylindrical bottles each of radius 3cm and height 9cm. How many bottles are required to empty the bowl?

Answer 1

Hint: Use the formula for the volume of a hemisphere to calculate the total volume of the fruit juice inside the hemisphere. Now calculate the volume of juice that can be filled in each bottle. Hence calculate the total number of bottles that can be filed by the fruit juice inside the hemisphere.

Complete step-by-step answer:

We shall first calculate the volume of the hemisphere.
We have r = radius of hemisphere = 18cm.
We know that volume of a hemisphere $=\dfrac{2}{3}\pi {{r}^{3}}$
Substituting r = 18, we get
Volume of hemisphere $=\dfrac{2}{3}\pi {{\left( 18 \right)}^{3}}=3888\pi c{{m}^{3}}$
Now we will calculate the volume of each bottle
We have r= radius of the base of cylinder = 3cm and h = height of cylinder = 9cm.
We know that volume of a cylinder $=\pi {{r}^{2}}h$.
Substituting r = 3 and h = 8, we get
The volume of the cylindrical bottle $=\pi {{\left( 3 \right)}^{2}}\left( 9 \right)=81\pi $
Let the number of bottles required to completely empty the hemisphere be x.
Then we have
 $\begin{align}
  & 3888\pi c{{m}^{3}}\le 81\pi x \\
 & \Rightarrow x\ge \dfrac{3888}{81}=48 \\
\end{align}$
Hence the minimum number of bottles required to empty the bowl = 48.

Note: [1] If we got the relation $x\ge 49.3$ instead, then the minimum number of bottles required = 49+1 = 50. This is because the number of bottles is an integer and hence if $x\ge 49.3$, means that we will require 50 bottles, but the 50th bottle will not be completely filled.
[2] When asked to find the required number of bottles, we find the minimum number of bottles required.