Problem 66 - Calculus Single Variable

Question

Physical therapists recommend that ramps for people who use wheelchairs rise not more than 1 in. for each foot of forward motion. Formulate this recommendation in terms of slope. If the entrance to a certain public building is \(30 \mathrm{ft}\) from the sidewalk and the front door is \(8 \mathrm{ft}\) off the ground (up a steep flight of stairs), how can a suitable wheelchair ramp be built?

Step-by-Step Solution

Answer

The ramp needs to be 96 feet long to meet the recommended maximum slope.

Step 1: Understanding Slope

The slope is defined as the 'rise over run', which in this context means the vertical rise (height) divided by the horizontal run (distance forward). The recommendation of a 1 inch rise for each foot forward translates to a slope of \( \frac{1}{12} \) since there are 12 inches in a foot.

Step 2: Calculate Requirement Slope for Ramp

Convert the 1 inch per foot recommendation into feet: \( \frac{1}{12} \) foot rise for each 1 foot of run. This is the maximum allowable slope for a wheelchair ramp as per physical therapy guidelines.

Step 3: Determine Ramp's Slope

For the given building, the entrance is 8 feet above ground level and 30 feet away from the sidewalk. Calculate the required slope as \( \frac{8}{30} \), where 8 feet is the vertical rise and 30 feet is the horizontal distance.

Step 4: Compare Required Slope with Recommended Slope

The calculated slope, \( \frac{8}{30} \), simplifies to approximately \( 0.267 \). The recommended maximum slope is \( \frac{1}{12} \) (approximately 0.0833). The required slope is steeper than recommended.

Step 5: Determine Necessary Adjustments

To satisfy the recommendation, the horizontal distance must be increased to decrease the slope. Set up the equation \( \frac{8}{x} = \frac{1}{12} \) to solve for \( x \), resulting in \( x = 96 \) feet. This means the ramp should extend 96 feet horizontally.

Key Concepts

Wheelchair Ramp

A wheelchair ramp is a specially built sloped passageway designed to replace stairs, enabling people using wheelchairs, strollers, or other mobility aids to access buildings safely. The main purpose of wheelchair ramps is to ensure accessibility and eliminate barriers for people with mobility challenges.
Ramps can be built into the structure of a building or added as a temporary installation. It's crucial for ramps to meet specific guidelines to provide adequate support and maintain safety.
Different organizations have set forth guidelines, but they all revolve around the core principle of creating a gradient that is both safe and comfortable for users.

Rise Over Run

The term 'rise over run' is a fundamental concept in understanding slopes. In the context of ramps, it refers to the vertical increase (rise) divided by the horizontal distance (run). This ratio is crucial in determining how steep a ramp will be. A gentler slope is achieved by having a smaller rise compared to the run, ensuring the ramp is easier to navigate.
Mathematically, a slope is expressed as:
\[ ext{slope} = \frac{\text{rise}}{\text{run}} \]For example, a 1 inch rise for each foot of horizontal distance translates to a slope of \( \frac{1}{12} \) when the rise is converted entirely into feet.
This means that for every 12 feet of horizontal run, the ramp rises by 1 foot.

Physical Therapy Guidelines

Physical therapy guidelines play a critical role when constructing wheelchair ramps. These guidelines are designed to ensure the safety and ease of use for individuals using mobility devices. In this context, the guidelines recommend a slope of no more than \( \frac{1}{12} \).
This ensures that the ramp isn’t too steep for safe navigation. A steep ramp can pose risks, including difficulties going uphill and the danger of losing control when moving downhill.

• A shallower slope reduces the exertion needed for ascent and descent.
• Adequate landings are also recommended to facilitate rest for users along lengthy ramps.
• Rails and non-slip surfaces are essential features for additional safety.

Adhering to these guidelines ensures accessibility while minimizing risk and promoting safety.

Ramp Construction

Constructing a wheelchair ramp requires careful planning and accurate calculations to meet guidelines and ensure safety. Consider the ramp for a building entrance 8 feet above ground, which is 30 feet from the sidewalk.
Initially, calculating the slope as \( \frac{8}{30} \) showed it to be too steep. To meet the prescribed \( \frac{1}{12} \) slope, adjustments involve extending the horizontal distance of the ramp to 96 feet:
\[ \frac{8}{96} = \frac{1}{12} \]This adaption ensures that the ramp is accessible and safe. Construction must also comply with local codes and regulations for length, width, and supporting structures.

• Materials should be durable and capable of withstanding weather conditions.
• The surface should have a non-slip coating to prevent accidents.
• Proper drainage systems can help handle water runoff.

These considerations ensure longevity, safety, and functionality of the ramp over time.