Problem 9 - Advanced Water Mathematics

Question

A contractor just installed 350 feet of 8 " diameter pipe. They want to wrap a corrosion resistant sleeve around the pipe and fill the pipe to pressure test it. How many gallons of water will the pipe hold and how many square feet of corrosion resistant sleeve are required to cover the whole pipe?

Step-by-Step Solution

Answer

The pipe will hold approximately 2,741.22 gallons of water, and 734.23 square feet of corrosion resistant sleeve are required.

Step 1: Calculate the Volume of the Pipe

To find the volume of the pipe, use the formula for the volume of a cylinder: \[ V = \frac{\text{πd}^2}{4}h \], where \( d = 8 \text{ inches} \) and \( h = 350 \text{ feet} \). First, convert the length to inches: \( h = 350 \text{ feet} \times 12 = 4200 \text{ inches} \). Now, calculate the volume: \[ V = \frac{\text{π} \times (8 \text{ inches})^2}{4} \times 4200 \text{ inches} \] \[ V = \frac{\text{π} \times 64}{4} \times 4200 = 201,600 \text{π} \text{ cubic inches} \]

Step 2: Convert Volume to Gallons

Convert the volume from cubic inches to gallons using the conversion factor: \( 1 \text{ gallon} = 231 \text{ cubic inches} \). \[ V_{\text{gallons}} = \frac{201,600 \text{π}}{231} \approx 2,741.22 \text{ gallons} \]

Step 3: Calculate the Lateral Surface Area of the Pipe

To find the lateral surface area, use the formula for the lateral surface area of a cylinder: \[ A = \text{πdh} \]. Here, \( d = 8 \text{ inches} \) and \( h = 4200 \text{ inches} \). Now, calculate the area: \[ A = \text{π} \times 8 \times 4200 = 33,600 \text{π} \text{ square inches} \] Convert the area to square feet by dividing by 144: \[ A_{\text{square feet}} = \frac{33,600 \text{π}}{144} \approx 734.23 \text{ square feet} \]

Key Concepts

Cylindrical Volume Calculation

Understanding how to calculate the volume of a cylinder is essential for solving many real-world problems. A cylinder's volume tells us how much space is inside the shape. To find the volume (\text{V}), you need to know two key pieces of information: the diameter of the base (\text{d}) and the height (\text{h)}. The formula to use is: \[ V = \frac{\text{πd}^2}{4}h \] The equation can be broken down into:

• \text{π} is the constant pi, approximately 3.14159.
• \text{d} is the diameter of the circular base.
• \text{h} is the height of the cylinder.

In our exercise, the diameter is 8 inches, and the height needs to be in the same units, so we convert 350 feet to inches, resulting in 4200 inches. Using the formula: \[ V = \frac{\text{π} \times (8 \text{ inches})^2}{4} \times 4200 \text{ inches} = 201,600 \text{π} \text{ cubic inches} \] Keep this value handy as we proceed to unit conversions.

Cylinder Surface Area

The surface area of a cylinder is crucial for determining how much material is needed to cover it. The formula incorporates the diameter and height again, focusing on the lateral surface area which is essentially the area around it: \[ A = \text{πdh} \] For our problem, \text{d} = 8 inches and \text{h} = 4200 inches. So, the surface area is: \[ A = \text{π} \times 8 \text{ inches} \times 4200 \text{ inches} = 33,600 \text{π} \text{ square inches} \] This result gives the area in square inches, but for practical purposes, especially large areas, converting this into square feet is often needed. Since 1 square foot is 144 square inches: \[ A_{\text{square feet}} = \frac{33,600 \text{π}}{144} \text{ square feet} \] Calculating this, we get approximately 734.23 square feet. This means to wrap the entire pipe, you need a sleeve that covers about 734.23 square feet.

Conversion Between Units

Converting units is a vital skill for handling different measures in mathematical problems. In this exercise, we dealt with converting height from feet to inches and volume from cubic inches to gallons. Here are the essential steps and conversions used:

• 1 foot = 12 inches. So, for height conversion: \[ 350 \text{ feet} \times 12 = 4200 \text{ inches} \]
• 1 gallon = 231 cubic inches.

When we calculated the volume of water the pipe can hold, we got a result in cubic inches (201,600π cubic inches). To find out how many gallons this converts to, we applied: \[ V_{\text{gallons}} = \frac{201,600 \text{π cubic inches}}{231 \text{ cubic inches per gallon}} \] This simplifies to approximately 2,741.22 gallons.
Remember: using correct conversions makes sure your solutions are accurate and meaningful.