Question

The cross-sectional area of a pipe is 25 square metres. What is the volumetric flow rate of the water if the velocity of the water is 10m/s?

Answer 1

Hint: Assume that the cross-sectional radius of the pipe is r. Use the fact that the area of the cross-section of the pipe or radius r is equal to $\pi r^2$. Hence find the radius of the pipe. Use the fact that the volume of the cylinder of radius r and height h is given by $V=\pi r^{2}h$. Hence determine the volumetric flow of water through the pipe.

Complete step-by-step answer:
Let the cross-sectional radius of the pipe be r.

We know that the area of a circle of radius r is given by $\pi r^2$
Since the cross-section of the pipe is circular, we have
$A=\pi r^2$, where A is the area of the cross-section of the pipe.
Since the cross-sectional area of the pipe is 25 square metres, we have
$\pi r^2=25$
Dividing both sides by $\pi$, we get
$r^2={\displaystyle \frac{25}{\pi }}$
Hence, we have
$r={\displaystyle \frac{5}{\sqrt{\pi }}}$
Length of the column of the water flowing through the pipe in 1 second = 10m(Since the flow rate is 10m/s)
We know that the volume of the cylinder of radius r and height h is given by $\pi r^{2}h$.
Hence, we have
The volume of the column of the water flowing through the pipe in 1 second $=\pi {\left({\displaystyle \frac{5}{\sqrt{\pi }}}\right)}^2\times 10=250m^3$
Hence, we have
The volumetric flow rate of water $=250m^3{s}^{-1}$

Note: Alternative Solution:
We have
The volume of a column of water of height h is given by
$$V=\pi r^{2}h=ar\left(\text{cross-section}\right)\times h$$
Hence, we have
The volumetric flow rate $=ar\left(\text{cross-section}\right)\times \text{velocity of water}$
Hence, we have
The volumetric flow rate $=25\times 10m^3{s}^{-1}=250m^3{s}^{-1}$, which is the same as obtained above.