Question

A rectangular water tank of base \[11m \times 6m\] contains water up to a height of 5m. If the water in the tank is transferred to a cylindrical tank of radius 3.5 m, find the height of the water level in the tank in cm.

Answer 1

Hint: We have been given the dimensions of the rectangular tank and the height of the water in it, which gives the volume of the water. So we will calculate the volume of water in the rectangular tank by using the formula \[\left( {{V_r}} \right) = L \times B \times H\] and substituting the values given in the question. As the same water is transferred to the cylindrical tank, the volume of the water will remain the same. We have been given the radius of the tank, but we have not been given the height, so we will assume height to be h. We can calculate the volume of water in the cylindrical tank by using the formula \[\left( {{V_c}} \right) = \pi \times {r^2} \times h\]. We will substitute the value of r given to us and thus we will then get the volume in terms of h. As we know the same amount of water is transferred from rectangular to cylindrical tank, volume of water will remain the same. As we have the value for volume of water in a rectangular tank and volume of water in a cylindrical tank in terms of h so we will equate the two volumes \[{V_c} = {V_r}\]. As all the other values are known except height h we will get the height of water in the cylinder. Now we have all the values in m, therefore the resulting value of h will also be in meter. But we have been asked to calculate the value of h in cm, so we will multiply the value of h in m by 100.

Complete step by step answer:
We will first refer to the figures given below for better understanding.

Let us first calculate the volume of water in the rectangular tank.
Length of rectangular tank = 11 m
Breadth of rectangular tank = 6 m
Height of rectangular tank = 5 m
We will substitute these values in the formula for volume of the rectangular tank which is as follows.
Volume of rectangular tank \[\left( {{V_r}} \right) = L \times B \times H\]
\[\Rightarrow {V_r} = 11 \times 6 \times 5\]
\[\Rightarrow {V_r} = 330\,{m^3}\] . . . . . . (1)
Now, let us calculate the volume of water in the cylindrical tank.
Radius of cylindrical tank = r = 3.5 m
Height of cylindrical tank = h
Now, we will use the formula for the volume of a cylindrical tank.
Volume of cylindrical tank \[\left( {{V_c}} \right) = \pi \times {r^2} \times h\]
Substituting the values in the formula for volume of cylindrical tank, we get
\[\Rightarrow {V_c} = \dfrac{{22}}{7} \times {\left( {3.5} \right)^2} \times h\]
\[\Rightarrow {V_c} = 38.5h\,\]. . . . . . (2)
As we know that the volume of water remains the same we can equate the equations (1) and (2).
On equating we get,
\[{V_c} = {V_r}\]
\[
  \Rightarrow 38.5h = 330 \\
\Rightarrow h = \dfrac{{330}}{{38.5}} \\
  \Rightarrow h = 8.57\,m \]
We get the height of the cylindrical tank as 8.57 m.
We have been asked to find the height in cm.
So to convert m to cm we will multiply the value of h in m by 100
\[
  h = 8.57 \times 100 \\
  \Rightarrow h = 857\,cm \]

Hence, the height of water level in a cylindrical tank is 857 cm.

Note: We need to keep in mind that as the same water from the rectangular tank is transferred to the cylindrical tank the volume of water remains same and hence we can equate the equations to find the height of water level in cylindrical tank. Also, keep in mind that all dimensions are given in m but we are asked to find the height of the water level in cm. To avoid too many calculations, it is best to first calculate the height in m and then convert into cm.