Question

The radius of a cylindrical vessel is 7 cm and its height is 12 cm. Two-third of the vessel is filled with water. A sphere having radius 6 cm is dropped into the water. Find the volume of the water that will come out of the vessel.
$
  {\text{A}}{\text{. }}196\pi \;{\text{c}}{{\text{m}}^3} \\
  {\text{B}}{\text{. 92}}\pi \;{\text{c}}{{\text{m}}^3} \\
  {\text{C}}{\text{. 288}}\pi \;{\text{c}}{{\text{m}}^3} \\
  {\text{D}}{\text{. 588}}\pi \;{\text{c}}{{\text{m}}^3} \\
 $

Answer 1

Hint- Here, we will be using the formulas for the volume of a cylinder having radius R and height H given by $\pi {{\text{R}}^2}{\text{H}}$ and the volume of a sphere of radius r given by $\dfrac{4}{3}\pi {r^3}$ in order to find the volumes required to evaluate the volume of the water that will come out of the vessel.

Complete step-by-step solution -

Given, Radius of the cylindrical vessel, R=7 cm
Height of the cylindrical vessel, H=12 cm
As we know that the volume of any cylinder with radius R and height H is $\pi {{\text{R}}^2}{\text{H}}$
Volume of the cylindrical vessel$ = \pi {\left( 7 \right)^2}\left( {12} \right) = 588\pi {\text{ c}}{{\text{m}}^3}$
It is also given that only two-third of this cylindrical vessel is filled with water.
So, the volume of the water present in the cylindrical vessel$ = \dfrac{2}{3} \times $Volume of the cylindrical vessel
$ \Rightarrow $Volume of the water present in the cylindrical vessel$ = \dfrac{2}{3} \times 588\pi = 392\pi {\text{ c}}{{\text{m}}^3}$
Now, a sphere of radius r=6 cm is dropped into the water present in the cylindrical vessel.
As we know that the volume of a sphere of radius r is $\dfrac{4}{3}\pi {r^3}$
$ \Rightarrow $Volume of the sphere dropped into the water$ = \dfrac{4}{3}\pi {\left( 6 \right)^3} = 288\pi {\text{ c}}{{\text{m}}^3}$

In order to find the total volume of the water that will come out of the vessel, we will subtract the volume of the cylindrical vessel from the total sum of the volume of water present in the vessel and the volume of the sphere dropped into the water.
Total volume of the water that will come out of the vessel = Volume of the water present in the cylindrical vessel + Volume of the sphere dropped into the water – Volume of the cylindrical vessel
$ \Rightarrow $ Total volume of the water that will come out of the vessel$ = 392\pi + 288\pi - 588\pi = 92\pi {\text{ c}}{{\text{m}}^3}$
Hence, option B is correct.

Note- In this particular problem, when a sphere is dropped into the water present in the cylindrical vessel then the level of the water will rise. The amount of the water that will come out of the cylindrical vessel will be the excess volume that is increased and extended beyond the volume of the cylindrical vessel.