Question

A farmer connects a pipe of internal diameter $ 20\,cm $ from a canal into a cylindrical tank in her field, which is $ 10\,m $ in diameter and $ 2\,m $ deep. If water flows through the pipe at the rate of $ 4\,\dfrac{{km}}{h} $ , in how much time will the tank be filled?

Answer 1

Hint: In this question we need to determine the time in which the tank will be filled. Here, we will determine the volume of pipe and the volume of the tank by using the volume of the cylinder formula, as both are in the shape of a cylinder. Then equate the volume of the pipe and volume of the tank to determine the height of the pipe. Then we will determine the time in which the tank will be filled when the rate of flow of water through the pipe is $ 4\,\dfrac{{km}}{h} $ .

Complete step-by-step answer:

First, let us determine the volume of pipe and the volume of the tank separately.
Now, let us determine the volume of pipe,
Pipe is in the form of a cylinder.
Therefore, we know that the volume of cylinder $ = \pi {r^2}h $
Here, it is given that the diameter of the pipe is $ 20\,cm $ .
Now, we know that, radius $ = \dfrac{{diameter}}{2} $
Therefore, $ r $ $ = \dfrac{{20}}{2} $
 $ = 10\,cm $
By converting $ cm $ into $ m $ , we have,
 $ = 10 \times \dfrac{1}{{100}}m $
 $ = \dfrac{1}{{10}}\,m $
Let the length of the pipe for filling the whole tank be $ h\,m $ .
Now, substituting the values in the volume of cylinder, we have,
 $ = \pi {\left( {\dfrac{1}{{10}}} \right)^2}h $
 $ = \pi \times \dfrac{1}{{100}} \times h $
 $ = \dfrac{{\pi h}}{{100}} $
Now, let us determine the volume of tank
Here, the tank is also in the form of a cylinder.
Therefore, we know that the volume of cylinder $ = \pi {r^2}h $
It is given that the cylindrical tank is $ 10\,m $ in diameter.
Now, we know that, radius $ = \dfrac{{diameter}}{2} $
\[r\] $ = \dfrac{{10}}{2}\,m $
 $ = \,5m $
It is also given that the cylindrical tank is $ 2\,m $ deep.
Therefore, $ h $ $ = 2\,m $
Substituting the values in the volume of a cylinder, we have,
 $ = \pi \times {\left( 5 \right)^2} \times 2 $
 $ = \pi \times 25 \times 2 $
 $ = 50\pi $
Now, volume of pipe=volume of tank
Therefore, $ \dfrac{{\pi h}}{{100}} = 50\pi $
 $ h = \dfrac{{50\pi \times 100}}{\pi } $
 $ h = 5000\,m $
 $ h = 5\,km $
It is given that water flows through the pipe at the rate of $ 4\,\dfrac{{km}}{h} $ .
We need to determine the time in which the tank will be filled.
Water flows in the pipe at the rate of $ 4km $ in $ 1\,hr $ .
So, water flows in the pipe at the rate of $ 1\,km $ $ = \dfrac{1}{4}\,hr $
Therefore, water flows in the pipe at the rate of $ 5\,km $ $ = \dfrac{5}{4}\,hr $
We know that $ 1hr = 60\min $ , therefore,
 $ = \dfrac{5}{4} \times 60\,\min $
 $ = 5 \times 15\,\min $
 $ = 75\,\min $
Hence in $ 75 $ minutes i.e., $ 1 $ hour $ 15 $ minutes, the tank will be filled.
So, the correct answer is “ $ 1 $ hour $ 15 $ minutes”.

Note: In this question, it is important to note here that we can also solve this question by determining the volume of water that flows in $ t $ hours from the pipe as \[area\,of\,cross\,section \times speed \times time\] and equating this with the volume of the tank, we can get the required time. While solving these types of questions, be clear with the formulas of the surface areas and the volumes.