Question

Water is flowing at the rate of 5km/hr through a pipe of diameter of 14cm into a rectangular tank which is 50m long and 44m wide. Determine the time in which the level of the water in the tank will rise by 7 cm.

Answer 1

Hint: We will use the Formula of Volume of Cylinder which is given by \[\pi {{r}^{2}}h\], where r is the radius of the base and h is the height of the cylinder and the Volume of Rectangular container which is given by V=lbh’, where l is length, b is breadth and h’ is height of the container.

Complete step-by-step answer:
Since the tank is rectangular in shape, we will find the volume of the rectangular tank which is given by the formula V=lbh’, where l represents length of the container,b represents the breadth of the container and h’ represents the height of the container.
Given in the question length l= 50m, breadth b= 44m and height h’=7cm
Since the height of the container is given in cm we change it into m so that all the values are in same unit, gives h’=0.07m
Now substituting the values of length l and breadth b and height h’ in the formula of volume we get,
\[V=(50)(44)(0.07)\]
\[\Rightarrow V=154.0{{m}^{3}}\]
Therefore, the volume of the container is 154m3.
Now we need to find the time in which the level of the water in the tank will rise by 7 cm. This would be obtained if we equate the volume of a rectangular tank when the height is h=0.07m and the volume of cylindrical pipe.
Equating both the volumes we get,
Volume of the rectangular = Volume of the cylinder
We know that the volume of the rectangular tank is 154 cubic meters, therefore substituting the volume of the rectangular tank in the above expression we get,
154 = Volume of the cylinder
We have the formula of volume of the cylinder as \[\pi {{r}^{2}}h\], where r is the radius of the base and h is the height of the cylinder
Applying the above formula we have the volume of the cylindrical pipe as \[\pi {{\left( \dfrac{d}{2} \right)}^{2}}(h)\] , where d is the diameter of the pipe and h is the required height
Putting radius of the base as \[r=\dfrac{d}{2}\] and equating both the expression we get,
\[154.0{{m}^{3}}=\pi {{\left( \dfrac{d}{2} \right)}^{2}}(h)\]
The diameter was given to be d= 14cm,changing it to meters we have=0.14 m
\[\begin{align}
  & \Rightarrow 154.0=\pi \left( \dfrac{0.14}{2} \right)(h) \\
 & \Rightarrow 154.0=\pi \left( 0.07 \right)(h) \\
 & \Rightarrow \dfrac{154}{\pi (0.07)}=h \\
 & \Rightarrow \dfrac{154(7)}{1.54}=h \\
 & \Rightarrow h=700 \\
\end{align}\]
So, the required height is 700m
Given in the question that the water travels 5000m in one hour and we need to determine the time required to fill this height h=700m
We have Time distance relation as
\[time=\dfrac{d}{s}\]
Here the distance travelled d is equal to the height h=700m and speed is s=5000m in 1 hour
\[\Rightarrow time=\dfrac{700}{5000}\] hour
\[\Rightarrow \]Time= 0.14hr
Now we convert the obtained hour into minutes
We have 1 hour = 60 min
\[\Rightarrow \]0.14 hour = (60)(0.14) min
\[\Rightarrow \]0.14 hour = 8.4 minutes
Therefore, time in which the level of the water in the tank will rise by 7 cm is 8.4 minutes

Note: Because the elements of the question for example the height, breadth etc. are given in different units, it is important to first convert all the given expressions and elements of the question in one single unit and then to proceed solving. As meters come under standard units, it will be better to convert all the given values in meters.