Problem 2 - Advanced Water Mathematics

Question

What is the volume in 2,000 feet of 18 -inch diameter pipe?

Step-by-Step Solution

Answer

3534.2 cubic feet.

Step 1 - Convert Diameter to Radius

The radius is half the diameter. Since the diameter is 18 inches, divide by 2 to get the radius. \[\text{Radius} = \frac{18 \text{ inches}}{2} = 9 \text{ inches}\]

Step 2 - Convert Radius to Feet

Convert the radius from inches to feet since the length of the pipe is in feet. 1 foot = 12 inches, so: \[\text{Radius} = \frac{9 \text{ inches}}{12 \text{ inches/foot}} = 0.75 \text{ feet}\]

Step 3 - Calculate the Cross-sectional Area

Use the formula for the area of a circle, which is \(A = \pi r^2\), where \(r\) is the radius. \[A = \pi (0.75 \text{ feet})^2 = \pi (0.5625 \text{ square feet}) = 1.7671 \text{ square feet} \]

Step 4 - Calculate the Volume of the Pipe

Multiply the cross-sectional area by the length of the pipe to get the volume. \[ V = A \times \text{Length} = 1.7671 \text{ square feet} \times 2000 \text{ feet} = 3534.2 \text{ cubic feet} \]

Key Concepts

Volume of a Cylinder

The volume of a cylinder can be found using the formula: \( V = \text{Base Area} \times \text{Height} \). For a cylinder, the base is a circle and its area can be calculated using the formula for the area of a circle, which is \(A = \text{π} r^2\). Here, 'r' is the radius of the cylinder's base.
The height or length of the cylinder is simply the distance between its two bases. In the context of pipes, this is the length of the pipe itself. Once you have the base area and height, multiplying them gives you the volume of the cylinder in cubic units.

Unit Conversion

In calculations, it's important to ensure all measurements are in the same unit system. Often you'll need to convert units to keep them consistent. For example, if the diameter of a pipe is given in inches but the length is in feet, you should convert either inches to feet or vice versa.
For converting inches to feet, remember that 1 foot equals 12 inches. Hence, if you have a measurement in inches, divide by 12 to convert it to feet. Conversely, multiply by 12 to convert feet to inches.

Cross-sectional Area Calculation

The cross-sectional area of a pipe is the area of the circle that forms its base. The formula to find the area of a circle is \( A = \text{π} r^2 \), with 'r' being the radius.
This area represents the amount of space within the pipe's base and is crucial for volume calculations. Once you have the radius (converted into the proper unit), squaring it and then multiplying by \( \text{π} \) (approximately 3.14159) will give you the cross-sectional area.

Diameter to Radius Conversion

The diameter of a circle is the distance across it, passing through the center. The radius is half the diameter. So, to convert the diameter to the radius, simply divide the diameter by 2.
For instance, if the diameter of a pipe is 18 inches, the radius would be:\[ \text{Radius} = \frac{18 \text{ inches}}{2} = 9 \text{ inches} \]. Converting from diameter to radius is fundamental because the area calculation requires the radius.

Pi in Calculations

Pi (π) is a mathematical constant representing the ratio of a circle's circumference to its diameter. It's approximately equal to 3.14159 and is often used in formulas involving circles.
In pipe volume calculations, π is used to find the cross-sectional area. When using π, you might use the approximate value 3.14 or the more precise 3.14159, depending on the required accuracy. Modern calculators and some software have a π key to ensure precision in calculations.