Problem 61 - Algebra for College Students

Question

Suppose that a highway rises a distance of 215 feet in a horizontal distance of 2640 feet. Express the grade of the highway to the nearest tenth of a percent.

Step-by-Step Solution

Answer

The grade of the highway is 8.1%.

Step 1: Understand the Concept of Grade

The grade of a highway is defined as the rise over run, expressed as a percentage. The grade indicates the steepness of the road.

Step 2: Calculate the Grade as a Ratio

To find the grade, divide the height that the highway rises by the horizontal distance. The formula for the grade is given by:\[ \text{Grade (as a ratio)} = \frac{\text{Rise}}{\text{Run}} = \frac{215}{2640} \]

Step 3: Calculate the Grade Percentage

Convert the grade from a ratio to a percentage by multiplying it by 100. \[ \text{Grade (percent)} = \left( \frac{215}{2640} \right) \times 100 \]

Step 4: Compute the Numerical Value

Calculate the numerical value:\[ \text{Grade (percent)} = \left( \frac{215}{2640} \right) \times 100 \approx 8.14 \]

Step 5: Round the Grade to the Nearest Tenth

Round the calculated decimal value of the grade to the nearest tenth of a percent.\[ 8.14 \approx 8.1 \]

Key Concepts

Algebra Ratios

Understanding algebra ratios is crucial for solving problems like highway grade calculation. A ratio is simply a comparison of two quantities. In our problem, the ratio describes how steep a segment of highway is compared to the flat road it covers.

When we talk about the ratio here, we're referring to the relationship between the rise of 215 feet and the run of 2640 feet. This is the initial step in computing the highway's grade. Formally, it is expressed as:

• Algebra Ratio = \( \frac{\text{Rise}}{\text{Run}} = \frac{215}{2640} \)

This ratio tells us that for every 2640 feet of horizontal distance, the highway rises by 215 feet. Remember, ratios are fundamental in algebra for comparing quantities to understand their relationships.

Percentage Conversion

Converting a ratio into a percentage helps to express the steepness of the highway in a more familiar form. Percentages are widely used to compare proportions easily because they are per one hundred.

To convert the ratio, \( \frac{215}{2640} \), to a percentage, we multiply by 100. This step is a straightforward multiplication:

• Percentage = \( \left( \frac{215}{2640} \right) \times 100 \)

This percentage conversion gives you the grade of the highway, which represents its incline as a part of 100. In our example, this works out to approximately 8.14 percent. This tells us that for every 100 feet horizontally, the road will rise approximately 8.14 feet.

Slope Measurement

Slope measurement is essential for understanding how steep a terrain or road is, particularly in construction and safety planning. While slope is often expressed as a fraction or ratio, converting it to a percentage makes it more understandable.

In our example, the slope is calculated using the formula for grade, which is the rise over the run. After converting this ratio to a percentage, it provides a clear picture of the highway's incline. Often, slope is required to be rounded off to a single decimal point for clarity and precision. For instance:

• Grade = approximately 8.14%
• Rounded Grade = 8.1%

This rounding process ensures that the slope measurement is practical for real-world applications. A slope is crucial for planning, so vehicles can navigate the highway safely without steep, unnavigable portions.