A conical vessel, with base radius 5 cm and height 24 cm, is full of water. This water is emptied into a cylindrical vessel of base radius 10 cm. Find the height to which the water will rise in the cylindrical vessel. (Use $\pi =\dfrac{22}{7}$)
Answer
363k+ views
Hint: Find the total volume of water using the formula for volume of a cone, given by $V=\dfrac{1}{3}\pi {{r}^{2}}h$, where base radius, r is 5 cm and height, h is 24 cm. Use the fact that the volume of water remains the same when it is emptied into a cylindrical vessel. For the volume of a cylindrical vessel use the formula $V=\pi {{r}^{2}}h$. Equate both the volumes to find the value of height of the cylinder.
Complete step-by-step answer:
We know that the volume of water in a vessel is the same as the volume of the vessel it is kept in. Thus, the total volume of water in the conical vessel can be calculated as the volume of the cone, given by $V=\dfrac{1}{3}\pi {{r}^{2}}h$. Using $r=5cm$ and $h=24cm$ in this formula, we get
$\begin{align}
& V=\dfrac{1}{3}\pi {{r}^{2}}h \\
& \Rightarrow V=\dfrac{1}{3}\pi {{\left( 5cm \right)}^{2}}\left( 24cm \right) \\
& \Rightarrow V=\pi \left( 25c{{m}^{2}} \right)\left( 8cm \right) \\
& \Rightarrow V=200\pi c{{m}^{3}} \\
\end{align}$
Now, since this entire volume is transferred to a cylindrical vessel, the volume of water would be the same as the volume of the cylinder, which can be given by $V=\pi {{r}^{2}}h$. This volume would be equal to the volume of the cube and the base radius is given as 10 cm. Equating the two volumes thus gives us
$\pi {{r}^{2}}h=200\pi c{{m}^{3}}$
Substituting the value of $r=10cm$ in this equation, we get
$\begin{align}
& \pi {{\left( 10cm \right)}^{2}}h=200\pi c{{m}^{3}} \\
& \Rightarrow \pi \left( 100c{{m}^{2}} \right)h=200\pi c{{m}^{3}} \\
& \Rightarrow 100h=200cm \\
& \Rightarrow h=2cm \\
\end{align}$
Thus the height upto which water is filled in the cylindrical vessel is 2 cm.
Note: To make calculations easier, the value of $\pi $ has not been substituted, even though it is given in the question, because $\pi $ occurs in the expression for both these volumes and hence, cancels out when the volumes are equated, thus reducing the calculations to a great extent.
Complete step-by-step answer:
We know that the volume of water in a vessel is the same as the volume of the vessel it is kept in. Thus, the total volume of water in the conical vessel can be calculated as the volume of the cone, given by $V=\dfrac{1}{3}\pi {{r}^{2}}h$. Using $r=5cm$ and $h=24cm$ in this formula, we get
$\begin{align}
& V=\dfrac{1}{3}\pi {{r}^{2}}h \\
& \Rightarrow V=\dfrac{1}{3}\pi {{\left( 5cm \right)}^{2}}\left( 24cm \right) \\
& \Rightarrow V=\pi \left( 25c{{m}^{2}} \right)\left( 8cm \right) \\
& \Rightarrow V=200\pi c{{m}^{3}} \\
\end{align}$
Now, since this entire volume is transferred to a cylindrical vessel, the volume of water would be the same as the volume of the cylinder, which can be given by $V=\pi {{r}^{2}}h$. This volume would be equal to the volume of the cube and the base radius is given as 10 cm. Equating the two volumes thus gives us
$\pi {{r}^{2}}h=200\pi c{{m}^{3}}$
Substituting the value of $r=10cm$ in this equation, we get
$\begin{align}
& \pi {{\left( 10cm \right)}^{2}}h=200\pi c{{m}^{3}} \\
& \Rightarrow \pi \left( 100c{{m}^{2}} \right)h=200\pi c{{m}^{3}} \\
& \Rightarrow 100h=200cm \\
& \Rightarrow h=2cm \\
\end{align}$
Thus the height upto which water is filled in the cylindrical vessel is 2 cm.
Note: To make calculations easier, the value of $\pi $ has not been substituted, even though it is given in the question, because $\pi $ occurs in the expression for both these volumes and hence, cancels out when the volumes are equated, thus reducing the calculations to a great extent.
Last updated date: 02nd Oct 2023
•
Total views: 363k
•
Views today: 4.63k
Recently Updated Pages
What do you mean by public facilities

Paragraph on Friendship

Slogan on Noise Pollution

Disadvantages of Advertising

Prepare a Pocket Guide on First Aid for your School

10 Slogans on Save the Tiger

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Who had given the title of Mahatma to Gandhi Ji A Bal class 10 social science CBSE

How many millions make a billion class 6 maths CBSE

Find the value of the expression given below sin 30circ class 11 maths CBSE

What is the past tense of read class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Number of Prime between 1 to 100 is class 6 maths CBSE

Who was the first President of the Indian National class 10 social science CBSE
