# Actually Applicable Application Problems and Brainteasers/Wheelchair Ramp Standards 2

## Overview

As mentioned in the original Wheelchair Ramp Standards problem listing, measuring the rise and run of real world ramps can be difficult. This is an example of a way around that issue, by using a protractor and the trigonometric tangent function. I'm not saying it's the best way, and it's definitely not the only way, but it's something I've come up with.

This diagram illustrates the geometric relationships involved. A right triangle illustrates a ramp. Its vertical side is labeled "opposite = rise" and its horizontal side is labeled "adjacent = run." The angle formed by the hypotenuse and horizontal sides is labeled, using an arrow, as "ramp angle." A pair of nested semicircles on the triangle's hypotenuse illustrates a protractor. A vertical line segment from the center of the semicircles (also touching the hypotenuse) illustrates a swingable arm used for reading angles from the protractor. The angle between this segment and the hypotenuse is labeled, using an angle, as "measured angle." At the top of the vertical segment there is a horizontal segment which is labeled, using an arrow, as "bubble level."

## General Method

### Part 1: Building the tool

#### Materials

• Bubble level
• Protractor with arm that swings around (used on paper for lining something up with the angle's edge)
• Something to attach them with (for example: rubber bands, glue, string, epoxy, etc.)

#### Recommended procedure

1. Make sure the protractor's arm is straight up. It must be very straight or the tool will not measure accurately.
• You may want to use a plumbline (a string with some kind of weight on the end) to check this.
2. Put the bubble level across the arm's end, laying down.
3. Adjust it until it is perfectly level (which should be perfectly perpendicular to the protractor's arm).
4. Attach it in place.
• Make sure it will stay attached without shifting or you will have to re-do building the tool to calibrate it every time you use it.

### Part 2: Measuring ramps

1. Put the protractor's back down against the ramp.
2. Swing the protractor's arm until the bubble level shows that it is level.
3. Read the protractor's angle by looking at where the swingable arm meets the curve.
4. Take the complement of the protractor's angle (that is, subtract the protractor's angle from 90°). This is the ramp's angle.
5. Take the tangent of the ramp's angle (that is, find the ratio of opposite to hypotenuse which corresponds to the ramp's angle).
• One way to calculate tangent is to divide sine by cosine.
6. As you can see form the diagram above, the opposite side of the triangle is the rise and the adjacent side of the triangle is the run, so the tangent ratio is the slope of the ramp. You can compare it with the standard, ${\frac {1}{12}}=0.08{\overline {3}}$ , to determine whether the ramp meets the accessibility standard.

## Problems

Is a ramp with an angle of 1° accessible?

Is a ramp with an angle of 45° accessible?

Is a ramp with an angle of 10° accessible?

Is a ramp with an angle of 5° accessible?

### Work Backwards

You can cut out steps 4 through 6 by figuring up once and for all what range of angle readings from step 3 will give an accessible result.