Problem 64 - Algebra for College Students

Question

Suppose that a county ordinance requires a \(2 \frac{1}{4} \%\) "fall" for a sewage pipe from the house to the main pipe at the street. How much vertical drop must there be for a horizontal distance of 45 feet? Express the answer to the nearest tenth of a foot.

Step-by-Step Solution

Answer

The vertical drop is approximately 1.0 feet.

Step 1: Understand the Problem

The problem discusses the "fall" of a sewage pipe, which is the vertical drop per unit horizontal distance. We're given a percentage and need to calculate the vertical drop for a horizontal distance of 45 feet.

Step 2: Convert the Percentage to a Decimal

The fall percentage is given as \(2 \frac{1}{4}\%\). First, convert this to a decimal by changing the mixed number to an improper fraction: \(2 \frac{1}{4} = \frac{9}{4}\). Then, convert the percentage to a decimal by dividing by 100: \(\frac{9}{4} \div 100 = 0.0225\).

Step 3: Calculate the Vertical Drop

Now, use the decimal fall percentage (0.0225) to find the vertical drop over 45 feet. Multiply the horizontal distance by the fall percentage: \(45 \times 0.0225 = 1.0125\).

Step 4: Round the Vertical Drop

Round the calculated vertical drop to the nearest tenth of a foot. The number 1.0125 rounded to the nearest tenth is 1.0.

Key Concepts

Understanding Percentage Calculations

Percentage calculations are essential in many real-life situations, including calculating interest rates, discounts, and as in our case, the slope or "fall" of a sewage pipe. A percentage is essentially a number or ratio expressed as a fraction of 100. In our problem, we are given a percentage that shows the fall of the pipe.
To deal with percentage calculations, you often need to convert percentages into decimals. This is done by dividing the percentage value by 100. For example, converting the percentage of the pipe’s fall from \(2 \frac{1}{4}\%\) into a decimal involves changing \(2 \frac{1}{4}\) into a fraction \(\frac{9}{4}\) and then dividing by 100 to get \(0.0225\).

• Convert mixed numbers to improper fractions when necessary.
• Always divide by 100 to go from percentage to decimal form.

Understanding this conversion is vital because it allows us to perform further calculations, such as finding the vertical distance over a horizontal stretch.

Calculating Vertical and Horizontal Distances

In geometry and physics, vertical and horizontal distances are commonly calculated to determine how far something changes position over a base distance. The problem of determining the "fall" of a pipe involves calculating how much vertical distance (i.e., drop) occurs for a specified horizontal distance.
The horizontal distance is the distance that runs parallel to the ground, and the vertical distance is the up or down movement perpendicular to the ground. When a percentage fall is given, like in our example, it indicates how much the ground should drop vertically over each unit of horizontal distance traveled.

• The horizontal distance in the example is fixed at 45 feet.
• The vertical drop is found by multiplying this horizontal distance by the fall percentage (converted to a decimal).

This mathematical approach can be applied to other scenarios where changes in elevation or slope need to be calculated.

Simplifying with Mathematical Conversions

Mathematical conversions often help us work seamlessly between different units and forms of numbers. In various problems, like the sewage pipe scenario, converting mixed numbers to improper fractions, and percentages to decimals, is a crucial step in simplifying and solving equations.
First, it’s significant to turn mixed numbers into improper fractions for easier division and multiplication, which is a common task in mathematical conversions.

• A mixed number is a whole number combined with a fraction, and it’s often more straightforward to convert it to an improper fraction for calculations.
• An improper fraction is one where the numerator is larger than the denominator, such as converting \(2 \frac{1}{4}\) to \(\frac{9}{4}\).
• From here, divide by 100 to convert a percentage to a decimal, since dividing by 100 shifts the decimal point two places to the left.

These conversions simplify multiplication and other computations needed to solve real-world problems, ensuring solutions are accurate and manageable.