Question

Water flows through a cylindrical pipe of internal diameter 7cm at 5m per second. Calculate:
i) The volume in litres, of water discharged by the pipe in 1 minute.
ii) The time in minutes, the pipe would take to fill an empty rectangular tank 4m by 3m by 2.31m.

Answer 1

Hint: Volume of the cylinder and cuboid can be given as $\pi {{r}^{2}}h$ and $\text{length}\times \text{breadth}\times \text{height}$ respectively, where r is the radius of cylinder and h is the height. Significance of the speed of water flow through the pipe is 5m per second. The water covers a 5m distance in 1 second in the pipe. So, try to calculate the volume of water in the pipe for 1 second and hence for 1 minute. Tank will get filled by pipe if water flow through the pipe is equal to the volume of the pipe.

Complete step-by-step answer:

We know that the capacity of any solid body is represented by its volume. So, water flowing through the pipe with an internal diameter of 7cm at 5m/s will represent the volume of the cylinder. Now, we have the speed of the flow of the water through the pipe is 5m/s. It means water will cover 5m distance in the pipe in 1 sec time duration. So, the volume of the water in 1 second through the pipe is given by calculating the volume of the cylinder with diameter 7cm and length 5m.

We know the volume of the cylinder is \[\pi {{r}^{2}}h\] where $\pi =\dfrac{22}{7}$ . r and h are the radius and height or length of the cylinder respectively. So, volume of water through pipe in 1 second can be given as volume of water through pipe in 1 sec $\pi {{r}^{2}}h.............\left( i \right)$
$\begin{align}
& \pi =\dfrac{22}{7} \\
& r=\dfrac{\text{diameter}}{2}=\dfrac{7}{2}cm \\
& h=5m \\
\end{align}$
We know 1m = 100cm, 5m = 500cm. So we get h = 500cm
Hence, we can put values of $\pi ,r,h$ in equation (i). so, we get volume through pipe in one second as
$\begin{align}
& =\pi {{r}^{2}}h \\
& =\dfrac{22}{7}\times {{\left( \dfrac{7}{2} \right)}^{2}}\times 500 \\
& =\dfrac{22}{7}\times \dfrac{49}{4}\times 500 \\
& =22\times 7\times 125 \\
& =19250c{{m}^{3}} \\
\end{align}$
Hence, volume through pipe in 1 second = $19250c{{m}^{3}}$ .
Now, coming to the question;
i) As we have already calculated the volume of water through the pipe in 1 second is $19250c{{m}^{3}}$ .
We know 1 minute = 60 seconds. So, the volume of water discharged in 1 second = $19250c{{m}^{3}}$ . Hence, volume of water discharge in 60 seconds $=60\times 19250c{{m}^{3}}=1155000c{{m}^{3}}$ .
We know that,
$\begin{align}
& 1L=1000c{{m}^{3}} \\
&1c{{m}^{3}}=\dfrac{1}{1000}L \\
&1155000c{{m}^{3}}=\dfrac{1}{1000}\times 1155000 \\
& =1155L \\
\end{align}$
Hence, the volume of water through the pipe in 1 minute is 1155 litres.

ii) let the time taken by the pipe to fill the rectangular tank be ‘t’ minutes. We know that the volume of the rectangular tank will be fulfilled by the pipe if the total volume of the water through the pipe in time ’t’ minutes is equal to the volume of the rectangular tank. So, we know the volume of the cuboid can be given by relation as
$\text{Volume of cuboid}=\text{length}\times \text{breadth}\times \text{height}$

Hence, volume of the tank can be given by the relation mentioned above. We know 1m = 100cm. Length of the tank = 4m = 400cm,
Breadth of the tank = 3m = 300cm,
Height of the tank = 2.31m = 231cm.
Hence, volume of the rectangular tank is,
$\begin{align}
& =400\times 300\times 231 \\
& =27720000c{{m}^{3}} \\
\end{align}$
Now, we the volume of water through pipe in 1 second is $19250c{{m}^{3}}.$ The volume through the pipe in t minutes is $t\times 60$ seconds can be given as
$\begin{align}
& t\times 60\times 19250c{{m}^{3}} \\
& =1155000tc{{m}^{3}} \\
\end{align}$
Now, we can equate the volume of the tank and volume of the water through the pipe in ‘t’ minutes. Hence we get
1155000t = 2770000
$t=\dfrac{27720}{1155}=24$
Hence, time taken to fulfil the rectangular tank is 24 minutes.

Note: Don’t confuse with the given speed of water, it represents the length of flow of water in one second. One may get confused while equating the volume of water in pipe in t minutes to the volume of the tank. Here, we are calculating the volume of water in 1 second and hence volume for ‘t’ minutes through it and we know the tank will get filled in ‘t’ minutes. Hence both should be equal as the quantity of water will remain the same. Unit conversion is also an important part. Always solve these types of problems only involving a single unit with length, time or anything else.