
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
594.6k+ views
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.
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.
Recently Updated Pages
In cricket, what is a "pink ball" primarily used for?

In cricket, what is the "new ball" phase?

In cricket, what is a "death over"?

What is the "Powerplay" in T20 cricket?

In cricket, what is a "super over"?

In cricket, what is a "tail-ender"?

Trending doubts
Why is there a time difference of about 5 hours between class 10 social science CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Write an application to the principal requesting five class 10 english CBSE

What is the median of the first 10 natural numbers class 10 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Who Won 36 Oscar Awards? Record Holder Revealed

