The Uniform Federal Accessibility Standards specify that the ramp angle used for a wheelchair ramp must be less than or equal to 4.78° . All ramps are created to be right triangles. The following diagram shows the parts of a ramp.

Question

The longest leg of a right triangle is labeled horizontal distance. The hypotenuse is labeled length of ramp. The angle opposite the short leg is labeled ramp angle. The short leg is labeled vertical rise.
If the length of the ramp is 16 inches and the vertical rise is 14 inches, what is a possible measurement for the horizontal distance?
If the ramp angle is 4.75° , determine the measure of the third angle of the triangle.
To meet the Uniform Federal Accessibility Standards, which side will be the shortest side of the triangle?
The length of a ramp is 3 feet long and its vertical rise is 20 inches. Explain why it is or is not possible to have a horizontal distance of 16 inches.

Answer 1

To solve the problems presented regarding the wheelchair ramp, let's break it down step by step.

1. Finding the Horizontal Distance: We know the ramp forms a right triangle, where the vertical rise is 14 inches and the length of the ramp (hypotenuse) is 16 inches. To find the horizontal distance, we can use the Pythagorean theorem:

   \[ \text{Length}^2 = \text{Vertical Rise}^2 + \text{Horizontal Distance}^2 \]

   Substituting the known values:

   \[ 16^2 = 14^2 + \text{Horizontal Distance}^2 \] \[ 256 = 196 + \text{Horizontal Distance}^2 \] \[ \text{Horizontal Distance}^2 = 256 - 196 = 60 \] \[ \text{Horizontal Distance} = \sqrt{60} \approx 7.75 \text{ inches} \]

   Thus, a possible measurement for the horizontal distance is approximately 7.75 inches.

2. Finding the Third Angle: In any triangle, the sum of the interior angles is \(180^\circ\). In a right triangle, one angle is \(90^\circ\), and the other two angles must add up to \(90^\circ\). Given that the ramp angle is \(4.75^\circ\), we can find the third angle:

   \[ \text{Third Angle} = 90^\circ - 4.75^\circ = 85.25^\circ \]

   Therefore, the measure of the third angle of the triangle is \(85.25^\circ\).

3. Identifying the Shortest Side: In any right triangle, the shortest side of the triangle is opposite the smallest angle. The angles we have are \(90^\circ\) (the right angle), \(4.75^\circ\) (the ramp angle), and \(85.25^\circ\) (the third angle). The smallest angle is \(4.75^\circ\), indicating that the side opposite this angle, which is the vertical rise, will be the shortest side. Thus, the shortest side of the triangle is the vertical rise.

4. Explaining the Horizontal Distance with Given Dimensions: In this case, we have a ramp length of 3 feet and a vertical rise of 20 inches. We need to determine if it's possible to have a horizontal distance of 16 inches.

   First, we should convert the length of the ramp from feet to inches. Since 3 feet is \(3 \times 12 = 36\) inches, we have:

   • Length of ramp = 36 inches
   • Vertical rise = 20 inches
   • We want to check if the horizontal distance can be 16 inches.

   Applying the Pythagorean theorem again:

   \[ \text{Length}^2 = \text{Vertical Rise}^2 + \text{Horizontal Distance}^2 \] \[ 36^2 = 20^2 + 16^2 \] \[ 1296 = 400 + 256 \] \[ 1296 = 656 \quad (\text{This is not true}) \]

   Since \(36^2\) does not equal the sum of \(20^2\) and \(16^2\), this indicates that it is not possible to have a horizontal distance of 16 inches with a ramp length of 36 inches and a vertical rise of 20 inches.

In conclusion:

• The horizontal distance is approximately 7.75 inches.
• The third angle is 85.25°.
• The shortest side is the vertical rise.
• It is not possible to have a horizontal distance of 16 inches with the given ramp dimensions.