Pipes, wells and tanks

Water in a pipe

Turn a tap on. Water pours out of the pipe. In one second, each drop moves forward a distance equal to the speed of the water. The water that comes out in that one second fills a cylinder. The cylinder has the radius of the pipe and a length equal to the speed.

So if the water moves at speed $v$ through a pipe of radius $r$,

Take care with the units. If $r$ is in centimetres and $v$ is in metres per second, change one of them first. A good plan is to write everything in centimetres, work out the volume in $\text{cm}^3$ and then divide by $1000$ for litres.

Filling a tank. The tank gets the volume of water that the pipe delivers. So

Example 1. A pipe of radius $3.5\ \text{cm}$ delivers water at $1\ \text{m/s}$ into a cylindrical tank of radius $70\ \text{cm}$. How long does it take to raise the water level by $1.5\ \text{m}$? Take $\pi = \frac{22}{7}$.

1. Water per second. $\dfrac{22}{7} \times (3.5\ \text{cm})^2 \times 100\ \text{cm} = 3850\ \text{cm}^3$.
2. Water needed. $\dfrac{22}{7} \times (70\ \text{cm})^2 \times 150\ \text{cm} = 2\,310\,000\ \text{cm}^3$.
3. Time. $2\,310\,000 \div 3850 = 600\ \text{s}$, which is $10$ minutes.

Earth from a well

A well is a cylinder dug into the ground. The earth that comes out has the same volume as the well, $\pi r^2 d$ for a well of radius $r$ and depth $d$. If this earth is spread evenly over a flat area $A$, the layer has a height

The earth may be spread over a rectangle, a circle or a ring round the well. For a ring with an inner radius $r$ and an outer radius $R$, the area is $\pi R^2 - \pi r^2$. In school problems we ignore the fact that loose earth takes more room than packed earth.

Example 2. A well of diameter $7\ \text{m}$ is dug $20\ \text{m}$ deep. The earth is spread evenly over a rectangular plot $35\ \text{m}$ by $22\ \text{m}$. How high is the earth layer? Take $\pi = \frac{22}{7}$.

1. The earth. $\dfrac{22}{7} \times (3.5\ \text{m})^2 \times 20\ \text{m} = 770\ \text{m}^3$.
2. The plot. $35\ \text{m} \times 22\ \text{m} = 770\ \text{m}^2$.
3. The height. $770\ \text{m}^3 \div 770\ \text{m}^2 = 1\ \text{m}$.

Other flows and rises

• Rain. The volume of rain on a roof is the area times the depth. One millimetre of rain on one square metre is exactly one litre.
• A solid dropped in water. If a solid goes completely under water, the water rises. The volume of water that moves is the volume of the solid. For a cylinder of radius $R$, the rise is $\dfrac{\text{volume of the solid}}{\pi R^2}$.

Example 3. A roof is $15\ \text{m}$ by $12\ \text{m}$. It rains $2\ \text{cm}$ in one hour. How many litres fall on the roof?

1. The area. $15\ \text{m} \times 12\ \text{m} = 180\ \text{m}^2$.
2. The depth. $2\ \text{cm} = 0.02\ \text{m}$.
3. The volume. $180\ \text{m}^2 \times 0.02\ \text{m} = 3.6\ \text{m}^3$, which is $3600$ litres.

Practise with new numbers

Try it in Python

This program finds how much water a pipe delivers, and how much it delivers each second, from the radius of the pipe and the speed of the water.

Common mistakes

• Using the diameter of the pipe as the radius. Halve the diameter first.
• Mixing centimetres and metres. Convert before you multiply.
• Forgetting the time. The pipe gives $\pi r^2 v$ each second, not in total.
• Using the area of the well's side for the earth. The earth is the volume of the well, $\pi r^2 d$.

Remember these

• Water from a pipe in time $t$ is $\pi r^2 v t$.
• The time to fill a tank is its volume divided by the volume per second.
• The earth from a well has the volume $\pi r^2 d$, and a layer of area $A$ has height volume divided by $A$.
• A solid under water raises the level by its volume divided by the area of the surface of the water.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Water leaves a pipe of cross-section $20\ \text{cm}^2$ at $50\ \text{cm/s}$. How many cubic centimetres leave in $10\ \text{s}$?

2. A tank holds $12\,000$ litres. A pipe fills it at $20$ litres every second. How long does it take?

3. Earth of volume $90\ \text{m}^3$ is spread over a rectangle $10\ \text{m}$ by $6\ \text{m}$. How high is the layer?

4. A cuboid of volume $240\ \text{cm}^3$ is dropped into a tank with a base of $120\ \text{cm}^2$ and sinks completely. By how much does the water rise?

5. A pipe has diameter $14\ \text{cm}$. What radius goes into the formula $\pi r^2 v t$?

References

• NCERT, Mathematics for Class 10, the chapter on Surface Areas and Volumes, for volumes of combinations of solids and conversion of a solid from one shape to another.
• Cambridge IGCSE Mathematics (0580), syllabus topic Mensuration, for volumes of cylinders, cones and spheres.