Question

A cylindrical tank has a radius of 154 cm. It is filled with water till a height of $3$ metres. If some water to a height of 4.5 metres is poured into it, what will be the increase in the volume of water?

Answer 1

Hint: Here we have to calculate the increase in the volume of water, so we have to use the formula for volume of cylinder, $v=\pi {{r}^{2}}(H-h)$ where $h$, is the initial height of water filled in the jar and $H$ is the final height of water filled. We have the values of $h=3m$ and $H=4.5m$.

Complete step-by-step solution -
Here, we have a cylindrical tank with radius, $r=154cm$ which is filled with water till a height of 3 metres as in the figure:

First we have to convert cm into m, we know that $1m=100cm$. Therefore, we can say that,
$\begin{align}
  & 154cm=\dfrac{154}{100}m \\
 & 154cm=1.54m \\
\end{align}$
That is, we got $r=1.54m$
Initially, the height of the cylinder, $h=3m$.
After that we add water till it reaches a height, $H=4.5m$
So, here we have to calculate the increase in the volume of water.
We know that the formula for the volume of cylinder, $V$is:
$V=\pi {{r}^{2}}h$, where $r$ is the radius of the cylinder and $h$ is the height of the cylinder.
Here, for the height $h$, the volume of the cylinder denoted as ${{V}_{1}}$ is given by,
${{V}_{1}}=\pi {{r}^{2}}h$
Similarly, for the height $H$, the volume of the cylinder denoted by ${{V}_{2}}$ is given by,
${{V}_{2}}=\pi {{r}^{2}}H$
Hence, the increase in the volume of water, denoted by $v$ is given by,
$\begin{align}
  & v={{V}_{2}}-{{V}_{1}} \\
 & v=\pi {{r}^{2}}H-\pi {{r}^{2}}h \\
\end{align}$
Now, by taking $\pi {{r}^{2}}$ outside we get:
$v=\pi {{r}^{2}}(H-h)$
In the next step, we have to substitute the values for $r=1.54,H=4$ and $h=3$. Thus, we get the equation:
$\begin{align}
  & v=\pi \times {{(1.54)}^{2}}(4.5-3) \\
 & v=\dfrac{22}{7}\times 1.54\times 1.54\times 1.5\text{ }....\text{ }\left( \pi =\dfrac{22}{7} \right) \\
\end{align}$
By cancellation we obtain:
$\begin{align}
  & v=22\times 0.22\times 1.54\times 1.5 \\
 & v=4.84\times 1.54\times 1.5 \\
 & v=7.4536\times 1.5 \\
 & v=11.1804 \\
\end{align}$
Hence, the increase in the volume of water is $11.1804{{m}^{3}}$

Note: Here, we have to calculate the increase in the volume of water, i.e. we have to subtract the volume of cylinder with height h from the volume of cylinder with height H. Since, it is increase in volume, don’t get confused that we have to add two volumes, it may lead to a wrong answer.