# 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 $\pi =\dfrac{22}{7}$)

Last updated date: 26th Mar 2023

•

Total views: 307.2k

•

Views today: 7.83k

Answer

Verified

307.2k+ 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.

Recently Updated Pages

If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

The coordinates of the points A and B are a0 and a0 class 11 maths JEE_Main

Trending doubts

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

Tropic of Cancer passes through how many states? Name them.

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

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE

What is per capita income

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

A Short Paragraph on our Country India