Problem 54 - Algebra A Combined Function

Question

Professional plumbers suggest that a sewer pipe should rise 0.25 inch for every horizontal foot. Find the recommended slope for a sewer pipe and write the slope as a grade, or percent. Round to the nearest percent.

Step-by-Step Solution

Answer

The recommended slope for a sewer pipe is 25%.

Step 1: Understanding the Slope Concept

The slope represents the rise over the run. In this scenario, the sewer pipe rises 0.25 inches for every horizontal foot. This means the slope is the ratio of vertical increase (rise) over a horizontal distance (run).

Step 2: Calculating the Slope as a Ratio

The given rise is 0.25 inches, and the run is 1 foot. Therefore, the slope is calculated by the formula: \[ \text{Slope} = \frac{\text{Rise}}{\text{Run}} = \frac{0.25}{1} \]Thus, the slope is 0.25.

Step 3: Converting Slope to a Percentage

To express the slope as a percentage, multiply the slope (ratio) by 100. \[ \text{Slope (as a percentage)} = 0.25 \times 100 = 25\% \]

Step 4: Rounding the Percentage to the Nearest Whole Number

Since 25% is already a whole number, there is no need for rounding. Therefore, the recommended grade for the sewer pipe is 25%.

Key Concepts

Rise Over Run

The concept of "rise over run" is foundational in understanding slopes. It is a simple way to express how much a surface inclines over a specific distance. The "rise" is the vertical change, and the "run" is the horizontal change.
For example, consider a sewer pipe that needs a slope. If it rises 0.25 inches for every foot horizontally, the rise over run ratio is 0.25 inches/foot.

• This ratio is used in various applications, from construction to mathematics.
• It helps in explaining the angle of inclination or steepness.

The formula to find the slope is straightforward: \[ \text{Slope} = \frac{\text{rise}}{\text{run}} \]This formula gives a slope in terms of a ratio. It is often easier for those working with angles to convert this ratio into a percentage or a decimal.

Percent Grade

The percent grade is a common and practical way to represent the steepness of a surface. It is especially useful in fields like engineering and transportation.
To convert a slope ratio into a percent grade, multiply the ratio by 100.

• This conversion gives the steepness as a percentage of the horizontal distance.
• It indicates how many units the rise is for every 100 units run.

For example, a sewer pipe with a rise of 0.25 inches per foot has a slope of 0.25. The conversion to percent grade is:\[ \text{Percent Grade} = 0.25 \times 100 = 25\%\]Percent grade provides a more intuitive understanding compared to a simple ratio. It allows for easy comparison of slopes and is crucial for designing infrastructure that needs precise incline measurements.

Sewer Pipe Gradient

In plumbing, the sewer pipe gradient is paramount for ensuring effective drainage. A proper slope ensures liquid flow moves with gravity, preventing main blockages.

• Sewer systems typically recommend specific slopes to handle waste efficiently.
• Too steep a slope can lead to liquids flowing too fast, leaving solids behind.
• Too shallow a slope might not allow adequate drainage, leading to backups.

Plumbers typically aim for gradients such as the recommendation of 0.25 inches per foot (or a 25% grade) for certain pipes.
This precise measurement facilitates:

• Consistent and reliable waste disposal.
• Minimized risk of clogging over time.

With an easy-to-understand gradient requirement, construction teams can ensure they build functional and reliable plumbing systems.