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Chapter 3: Problem 8

A wheelchair service ramp: The Americans with Disabilities Act (ADA) requires, among other things, that wheelchair service ramps have a slope not exceeding \(\frac{1}{12}\). a. Suppose the front steps of a building are 2 feet high. You want to make a ramp conforming to ADA standards that reaches from the ground to the top of the steps. How far away from the building is the base of the ramp? b. Another way to give specifications on a ramp is to give allowable inches of rise per foot of run. In these terms, how many inches of rise does the ADA requirement allow in 1 foot of run?

Short Answer

Expert verified
a. 24 feet. b. 1 inch per foot.

Step by step solution

01

Understand Slope Requirement

The ADA requires that the slope of the ramp does not exceed \( \frac{1}{12} \). This means that for every 1 unit of vertical rise, there should be at least 12 units of horizontal run.
02

Solve Part A - Setup the Equation

We know the height of the ramp is 2 feet. According to the slope requirement, the ratio of the rise (height) to the run (base) is given by \( \frac{1}{12} \). Thus, we can set up the following equation for the run (\( R \)): \[ \frac{2}{R} = \frac{1}{12} \].
03

Solve Part A - Calculate the Run

Cross-multiply to find \( R \): \[ 2 \times 12 = 1 \times R \] leading to \( R = 24 \). Therefore, the base of the ramp should be 24 feet away from the building.
04

Solve Part B - Convert Slope to Inches per Foot

To find how many inches of rise are allowed per foot of run, first note that 1 foot is 12 inches. Since the slope is \( \frac{1}{12} \), for each 12 inches (1 foot) of horizontal run, there is 1 unit of rise. Thus, in inches, the rise is \( 1 \) inch per 12 inches of run.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope Requirements
When designing a ramp, understanding slope requirements is crucial to ensure safety and compliance.
The slope of a ramp is essentially a measure of how steep it is, calculated as the vertical rise divided by the horizontal run. For ADA compliance, ramps must not exceed a slope of \(\frac{1}{12}\). This means for every foot the ramp rises, it must run at least 12 feet horizontally.

Why is this slope important?
  • Safety: A steeper slope could make the ramp unsafe for wheelchair users.
  • Accessibility: Easier slopes ensure more individuals can comfortably use the ramp.
  • Compliance: Meeting these regulations helps avoid legal issues.
To visualize, think about climbing a gentle slope versus a steep hill鈥攇entle slopes are easier and safer.
Wheelchair Accessibility
Ensuring wheelchair accessibility involves more than just installing a ramp. It considers all the features that make a building or space usable for individuals who rely on wheelchairs.
ADA guidelines set out clear requirements for ramps to ensure they are safe and usable for everyone.

Key aspects include:
  • Ramp Slope: As mentioned, the slope should not exceed \(\frac{1}{12}\) to be gentle enough for users.
  • Width: Ramps must be wide enough to accommodate wheelchairs, providing a minimum of 36 inches of clear width.
  • Landings: There should be level landings at the top and bottom of the ramp, and for every 30 feet of ramp, to provide rest areas.
  • Surface: The surface should be non-slip to prevent accidents.
Considering these guidelines helps create a more inclusive environment, enabling access for people with disabilities.
Educational Problem Solving
Educational problem-solving in the context of ADA ramp specifications challenges students to apply mathematical thinking to real-world scenarios.
These exercises teach students how to:
  • Understand and interpret regulations, taking abstract numbers and converting them into practical applications.
  • Calculate slope and dimensions based on height requirements, fostering skills in setting up and solving equations.
  • Analyze accessibility needs, encouraging critical thinking about usability and safety.
By solving these types of problems, students enhance their ability to tackle technical problems, preparing them for future engineering, architecture, or planning roles that consider accessibility.
Mathematical Modeling
Mathematical modeling involves creating a mathematical representation of a real-world scenario to predict and analyze outcomes.
In the case of ADA ramp specifications, the model is a simple linear model that uses slope as the key variable.

Here's how it's done:
  • Define the Parameters: Known variables include the rise (height of the ramp steps) and the slope regulation \(\frac{1}{12}\).
  • Create the Equation: Use these variables to set up an equation \(\frac{rise}{run} = \frac{1}{12}\) to represent the scenario.
  • Solve for Unknowns: Manipulate the equation to solve for the unknown, such as the horizontal run required.
  • Analyze the Result: Use the result to apply practically, ensuring the ramp will meet regulatory requirements.
This type of problem shows how understanding and simulating scenarios with mathematical equations can effectively guide practical decisions.

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Most popular questions from this chapter

Where lines with different slopes meet: On the same coordinate axes, draw one line with vertical intercept 2 and slope 3 and another with vertical intercept 4 and slope 1. Do these lines cross? If so, do they cross to the right or left of the vertical axis? In general, if one line has its vertical intercept below the vertical intercept of another, what conditions on the slope will ensure that the lines cross to the right of the vertical axis?

A line with given vertical intercept and slope: On coordinate axes, draw a line with vertical intercept 3 and slope 1. What is its horizontal intercept?

Lines with the same slope: On the same coordinate axes, draw two lines, each of slope 2 . The first line has vertical intercept 1 , and the second has vertical intercept 3. Do the lines cross? In general, what can you say about different lines with the same slope?

Mixing feed: A milling company wants to mix alfalfa, which contain \(20 \%\) protein, and wheat mids, \({ }^{28}\) which contain \(15 \%\) protein, to make cattle feed. a. If you make a mixture of 30 pounds of alfalfa and 40 pounds of wheat mids, how many pounds of protein are in the mixture? (Hint: In the 30 pounds of alfalfa there are \(30 \times 0.2=6\) pounds of protein.) b. Write a formula that gives the amount of protein in a mixture of \(a\) pounds of alfalfa and \(w\) pounds of wheat mids. c. Suppose the milling company wants to make 1000 pounds of cattle feed that contains \(17 \%\) protein. How many pounds of alfalfa and how many pounds of wheat mids must be used?

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