Problem 50 - Introductory and Intermediate Algebra for College Students 4th

Question

A college campus goes beyond the standards described in Exercise \(49 .\) All wheelchair ramps on campus are designed so that every vertical rise of 1 foot is accompanied by a horizontal run of 14 feet. What is the grade of such a ramp? Round to the nearest tenth of a percent.

Step-by-Step Solution

Answer

The grade of the ramp is approximately 7.1%.

Step 1: Understand what the grade of a ramp means

Grade of a ramp can be calculated as the ratio of the vertical rise to the horizontal run, and is typically expressed as a percentage. Here, we are given that for every vertical rise of 1 foot, there is a horizontal run of 14 feet.

Step 2: Calculate the grade of the ramp

To find the grade, we divide the vertical rise by the horizontal run and multiply by 100% to express as a percentage. Thus, grade (in %) = (1 foot / 14 feet) * 100%.

Step 3: Round to the nearest tenth

Once the calculation is done, the last step is rounding the result to the nearest tenth of a percent as requested by the problem.

Key Concepts

Ratio and Proportion

When we talk about the grade of a ramp, we are essentially discussing the concept of ratio and proportion. A ratio is a relationship between two quantities, expressing how much of one quantity there is relative to another. In this case, we are looking at the ratio of the vertical rise to the horizontal run.

To comprehend this in a real-world context, consider a ramp as a triangle. The vertical rise is the height, and the horizontal run is the base. Proportion comes into play when we compare this ratio to other ramps, or when standards dictate a certain slope for accessibility.

It's crucial to understand that ratios can be scaled. If a ramp is compliant with the ratio of 1 foot of rise to 14 feet of run, then a ramp with a 2-foot rise would need a 28-foot run to maintain the same proportion and, therefore, the same grade. This concept of keeping the same grade at different scales is a practical application of ratios and proportions in designing accessible spaces.

Percentages

Percentages are an integral part of many mathematical calculations, especially when expressing ratios in a more understandable format. In the context of our ramp, the grade is the vertical rise expressed as a percentage of the horizontal run.

What does this mean? A percentage is essentially a ratio where the second term is always 100. It represents a part per hundred, making it easier to visualize and compare. For our ramp's grade, we calculate the percentage by using the formula: \( \text{grade} (\%) = \left( \frac{\text{vertical rise}}{\text{horizontal run}} \right) \times 100% \).

The key here is that by converting a ratio into a percentage, we are normalizing the comparison across different contexts. Regardless of the actual dimensions of the ramp, we can understand and compare its steepness by this percentage.

Rounding Numbers

Rounding numbers is a mathematical technique used to simplify figures to make them easier to work with or communicate. In our ramp scenario, we are asked to round the grade to the nearest tenth of a percent.

This means that after calculating the exact percentage, we must adjust it to one decimal place. If the second decimal is five or above, we round up; if it's four or below, we round down. This rounding makes the value easier to read and reduces the level of precision to what's practically necessary.

In real-life situations, exact precision down to several decimal places is often unnecessary; a degree of approximation is acceptable. Rounding also aids in preventing the introduction of minor errors in subsequent calculations or in the interpretation of the number.