Question

A glass cylinder with diameter 20 cm has water to a height of 9 cm. A metal cube of 8
cm edge is immersed in it completely. Calculate the height by which water will rise in the
cylinder. Use $\pi =\dfrac{{22}}{7}\\$.

Answer 1

Hint: In this problem, first we have to find the volume of the cylinder. Volume of a cylinder can
be calculated by$\pi {r^2}h$, where $r$ is the radius, $h$ is the height of the cylinder. We have
to substitute the given value in the formula. This will give the volume of the cylinder. Like that,
we need to calculate the volume of the cube as well using the volume formula${a^3}$, where
$a$ is the edge length of the cube. Finally we have to compare the volume of the cylinder and the
volume of the cube.

Complete step-by-step answer:
Given that the diameter of the cylinder is $20\;{\rm{cm}}$ and the height is $9\;{\rm{cm}}$.
Converting the diameter to the radius.
Radius of the cylinder $\begin{array}{c} = \dfrac{{20}}{2}\\ = 10\;{\rm{cm}}\end{array}$

It is known that the volume of the cylinder is $\pi {r^2}h$, where $r$ is the radius and $h$ is the
height of the cylinder.
Step I
Let the height of the cylinder is $h$.
Substituting the value of 10 for $r$.
Volume $ = \dfrac{{22}}{7} \times {10^2} \times h$
$ = \dfrac{{2200h}}{7}\;{\rm{c}}{{\rm{m}}^3}$ (1)
Similarly, calculating the volume of the metal cube.
Step II
Known that the volume of a cube is ${a^3}$, where $a$ is the edge length.
Given that the edge length of the cube $ = 8\;{\rm{cm}}$
Substituting the value 8 for $a$. We get,
Volume $ = {8^3}$
$ = 512\;{\rm{c}}{{\rm{m}}^3}$
(2)
Step III
Now, we have to calculate the value of $h$.
Thus, comparing (1) and (2) and solving for $h$.
$\begin{array}{c}\dfrac{{2200h}}{7} = 512\\h = \dfrac{{512 \times 7}}{{2200}}\\h \approx
1.6\;{\rm{cm}}\end{array}$
Hence, the height by which the water rises in the cylinder is $1.6\;{\rm{cm}}$.

Note: Here we have to determine the height of the cylinder by which the water rises. In step I,
since the radius of the cylinder is given, so by substituting the value in the formula of the
cylinder we calculated the volume of the cylinder. In step II, similarly for the cube we calculated
the volume of the cube for the given edge length. Step III, given that both the volume of cylinder
and the cube is the same, so by comparing them we get the required height of the cylinder.