Calculate Water Pipe Surface Area & Volume - Ask an Expert

• MHB
• Monoxdifly
In summary: Lso the total surface area of water in contact with the pipe is $\boxed{2A+sL}$ where $L=240\,\mathrm{cm}$.
Monoxdifly
MHB
A water pipe whose diameter is 84 cm dan length is 2,4 m can contain rain water with the water height 68 cm like in the attached picture.
View attachment 8252
Determine:
a. The surface area which gets contact with the water
b. The volume of the water (in liters)

So... How do I do? What is the simple way to determine the area of a... truncated circle? (dunno what the proper term is)

Attachments

• Water Pipe.JPG
4.4 KB · Views: 95
How many boards have you posted this to? On "My Math Forum" I wrote

Looks like a pretty standard Calculus II problem. Start by drawing a cross section of the pipe on a coordinate system- a circle with center at (0, 0) and diameter 84, so radius 42, has equation $$\displaystyle x^2+ y^2= 42^2$$. The bottom of that circle is at y= -42, the top is at y= 42. The surface of the water is a horizontal line at y= 68- 42= 26. At every y, a horizontal line intersects the circle at $$\displaystyle x= \pm \sqrt{42^2- y^2}$$ so has length $$\displaystyle 2\sqrt{42^2- y^2}$$. Taking a 'thickness' of $$\displaystyle \Delta y$$, the area of that thin horizontal strip is $$\displaystyle 2\sqrt{42^2- y^2}\Delta y$$. Adding those gives the Riemann sum, $$\displaystyle \sigma 2\sqrt{42^2- y^2}\Delta y$$, that, in the limit, is the integral for the area, $$\displaystyle 2\int_{-42}^{26}\sqrt{42^2- y^2}dy$$.

Consider the following diagram:

View attachment 8253

To find the sum of the shaded areas, I would begin by finding the angle $\theta$ in terms of $h$ and $r$. Can you do that?

Attachments

• mhb_help_0001.png
4.2 KB · Views: 88
Country Boy said:
How many boards have you posted this to?
4 (MMF, MIF, MHB, and MHF).

Is there any approach without using calculus or trigonometry? This is supposed to be for 9 graders.

You don’t need calculus. I would use MarkFL’s approach as follows.

Find the angle $\theta$ (and make sure it’s in radians). The area of the yellow triangle in Mark’s diagram is $A_1=\frac12r^2\sin2\theta$. The area of the olive-green area below it is $A_2=\frac12r^2(2\pi-2\theta)$. So the area of the water in contact with one end of the pipe is $A=A_1+A_2$. Thus:

• The length of the major arc of the circle enclosing the olive-green area is $s=r(2\pi-2\theta)$. Hence the total surface area of water in contact with the pipe is $\boxed{2A+sL}$ where $L=240\,\mathrm{cm}$ is the length of the pipe. The answer here is in square centimetres.

• The volume of water is $\boxed{AL}$, which is in cubic centimetres so you’ll need to convert to litres: $1\,\mathrm{cm^3}\ =\ 0.001\,\ell$.

Monoxdifly said:
Is there any approach without using calculus or trigonometry? This is supposed to be for 9 graders.

We can do (a) without calculus or trigonometry.
Starting with MarkFL's picture we need to find the size at the top.
We can do that with Pythagoras can't we?
The triangle has hypotenuse $\frac 12\cdot 84 = 42\text{ cm}$ and vertical side $68 - \frac 12 \cdot 84 = 26\text{ cm}$ doesn't it?

(b) is a bit more complicated. I think we really need the angle $\theta$ in Mark's drawing.
Then we can find the fraction of a circle disk that corresponds to $360^\circ - 2\theta$.
And we can add the area of the two triangles, which we can find from the numbers we found in (a).
Multiply with the length of the cylinder and we have the volume.

I like Serena said:
We can do (a) without calculus or trigonometry.
Starting with MarkFL's picture we need to find the size at the top.
We can do that with Pythagoras can't we?
The triangle has hypotenuse $\frac 12\cdot 84 = 42\text{ cm}$ and vertical side $68 - \frac 12 \cdot 84 = 26\text{ cm}$ doesn't it?

But what about the surface area?

Monoxdifly said:
But what about the surface area?

You've got the two ends of the pipe already, and then along the pipe of length $$L$$ you have:

$$\displaystyle r(2\pi-2\theta)L$$

1. How do I calculate the surface area of a water pipe?

To calculate the surface area of a water pipe, you will need to measure the diameter and length of the pipe. Then, use the formula SA = 2πr(r+h), where r is the radius and h is the height or length of the pipe. This formula assumes that the pipe is circular in shape.

2. What is the formula for calculating the volume of a water pipe?

The formula for calculating the volume of a water pipe is V = πr2h, where r is the radius and h is the height or length of the pipe. This formula also assumes that the pipe is circular in shape.

3. Can I use the same formula for calculating the surface area and volume of any type of water pipe?

Yes, the formulas for calculating the surface area and volume of a water pipe can be used for any type of pipe, as long as it is circular in shape. However, if the pipe is not circular, you will need to use different formulas.

4. How do I convert the units of measurement for the surface area and volume of a water pipe?

If your measurements are in different units, you will need to convert them to the same unit before using the formulas. For example, if the diameter of the pipe is measured in inches, but the length is measured in feet, you will need to convert the diameter to feet before using the formulas. You can use a conversion calculator or multiply the measurements by the appropriate conversion factor.

5. What factors can affect the accuracy of the calculated surface area and volume of a water pipe?

The accuracy of the calculated surface area and volume of a water pipe can be affected by factors such as the precision of the measurements, the shape of the pipe, and any irregularities or bends in the pipe. It is important to take accurate measurements and account for any variations in the shape of the pipe to get the most accurate results.

Replies
7
Views
2K
Replies
6
Views
2K
Replies
5
Views
1K
Replies
9
Views
471
Replies
10
Views
3K
Replies
5
Views
745
Replies
2
Views
1K
Replies
7
Views
2K
Replies
3
Views
2K
Replies
4
Views
5K