Problem 95 - College Algebra and Calculus: An Applied Approach

Question

The maximum recommended slope of a wheelchair ramp is \(\frac{1}{12}\). A business is installing a wheelchair ramp that rises 34 inches over a horizontal length of 30 feet. Is the ramp steeper than recommended? (Source: Americans with Disabilities Act Handbook)

Step-by-Step Solution

Answer

Yes, the ramp is steeper than the recommended slope of \(\frac{1}{12}\).

Step 1: Convert all measurements to the same unit

The given measurements are in different units (inches and feet). To work with the measurements, they need to be in the same unit. We know that 1 foot is equal to 12 inches, so we can convert the horizontal length from feet to inches. The horizontal length in inches is therefore \(30 \times 12 = 360\) inches.

Step 2: Calculate the slope of the ramp

Now, let's calculate the slope of the ramp by dividing the rise (or vertical change) by the run (or horizontal change). So, the slope of the ramp is \(\frac{34}{360}\). This simplifies to approximately 0.094.

Step 3: Compare the actual slope with the recommended slope

The recommended slope is \(\frac{1}{12}\), which is approximately 0.083. As such, since 0.094 is greater than 0.083, we can conclude that the slope of the ramp is steeper than recommended.

Key Concepts

Slope Calculation

Understanding the concept of slope is essential, especially when you're dealing with structures like wheelchair ramps. Slope describes the steepness or incline of a surface. It is calculated using the formula: \[\text{slope} = \frac{\text{rise}}{\text{run}}\]Here "rise" refers to the vertical change, while "run" is the horizontal change. The slope is the ratio of these two values. When dealing with specific applications—like ramps—a gentle slope is usually preferred to ensure safety and accessibility, which is why it needs to be measured accurately. In this exercise, more than just calculating, the comparison of the obtained slope with a recommended value is crucial. This ensures that the ramp falls within safe and acceptable limits for its intended use.

Unit Conversion

Unit conversion is a vital step when you are working with dimensions. To compare or calculate the slope accurately, all measurements must be in the same units. In our example, the rise was given in inches and the horizontal run in feet. To convert feet to inches, we use the conversion factor:

• 1 foot = 12 inches

Therefore, 30 feet is converted to inches by multiplying it by 12, resulting in 360 inches. This makes it easier to work with and perform calculations without errors arising from differing units. Accurate unit conversion is not only crucial for solving textbook problems but also for real-world applications across various fields.

Americans with Disabilities Act

The Americans with Disabilities Act (ADA) sets numerous guidelines to ensure accessibility for individuals with disabilities. Regarding ramps, the ADA recommends a maximum slope of 1:12, which translates to a slope value of approximately 0.083 when expressed as a decimal. This guideline ensures that ramps are not too steep, thereby enabling individuals, particularly those using wheelchairs, to navigate them safely and comfortably. Violating these guidelines might not only result in a structure being impractical but also non-compliant with legal standards. Therefore, following these recommendations is crucial not only from a design perspective but also to adhere to legal obligations.