Question

Water flows at the rate of 10m/min through a cylindrical pipe of diameter 5mm. How long will it take to fill a conical vessel of diameter 40cm and depth 24cm?

Answer 1

- Hint: Assume that the time taken to fill the conical vessel is t minutes. Find the amount of water flowing through the pipe in 1 minute. Use the fact that the volume of a cylinder of radius r and height h is given by $V=\pi {{r}^{2}}h$. Hence determine the water flowing through the pipe in t minutes in terms of t. Use the fact that the volume of a cone of radius r and height h is given by $V=\dfrac{1}{3}\pi {{r}^{2}}h$. Hence find the volume of the conical vessel. Use the fact that the amount of water flowing through the pipe in t minutes is equal to the volume of the vessel. Hence form an equation in t. Solve for t and hence find the time taken to fill the conical vessel.


Complete step-by-step solution -

Let the time taken to fill the conical vessel be t minutes.
Length of the cylindrical column of water flowing through the pipe in 1 minute = 10m.
Radius of the cylindrical column of water = 2.5mm = 0.0025 m
We know that the volume of a cylinder of radius r and height h is given by $V=\pi {{r}^{2}}h$.
Hence, we have
The amount of water flowing through the pipe in 1-minute $=\pi {{r}^{2}}h=\pi \times \dfrac{2.5}{1000}\times \dfrac{2.5}{1000}\times 10=\dfrac{6.25\pi }{100000}$ cubic metres
Hence the amount of water flowing through the pipe in t minutes $=\dfrac{6.25\pi t}{100000}$
Now, we know that the volume of a cone of radius r and height h is given by $V=\dfrac{1}{3}\pi {{r}^{2}}h$.
Here diameter of the base of the conical vessel =$40cm=0.4m$
Hence, we have
The radius of the base of the conical vessel $=\dfrac{0.4}{2}=0.2m$
Also, we have
Height of the conical vessel $=20cm=0.2m$
Hence, we have
The volume of the conical vessel $=\dfrac{1}{3}\pi {{\left( 0.2 \right)}^{2}}\left( 0.24 \right)=\dfrac{96\pi }{30000}$ cubic metres
Since the amount of water flowing through the pipe in t minutes is equal to the volume of the vessel, we have
$\dfrac{6.25\pi t}{100000}=\dfrac{96\pi }{30000}$
Hence, we have
$\dfrac{6.25\pi t}{10}=\dfrac{96\pi }{3}$
Dividing both sides by $\pi $, we get
$\dfrac{6.25}{10}t=32$
Hence, we have
$t=\dfrac{320}{6.25}=51.2$
Hence the time taken to fill the vessel is 51.2 minutes.

Note: In mensuration problems students commit mistakes in taking care of the units of the quantities. It is important to keep the units of the terms into consideration while solving problems. Many students forget that and end up having incorrect results. This can be avoided by writing at the end of each quantity the units in which it is expressed.