Problem 54 - Calculus

Question

Find the angle, in degrees, that a wheelchair ramp makes with the ground if the ramp rises 1 foot over a horizontal distance of (a) \(12 \mathrm{ft},\) the normal requirement \(^{36}\) (b) \(8 \mathrm{ft},\) the steepest ramp legally permitted (c) \(20 \mathrm{ft}\), the recommendation if snow can be expected on the ramp

Step-by-Step Solution

Answer

(a) \(4.76^\circ\), (b) \(7.13^\circ\), (c) \(2.86^\circ\)

Step 1: Identify the Right Triangle

The situation with the wheelchair ramp forms a right triangle where the vertical rise is 1 foot, the hypotenuse is the ramp itself, and the horizontal distance is given for each case.

Step 2: Use Trigonometric Functions

To find the angle the ramp makes with the ground, we will use the tangent function, which is defined as the opposite side over the adjacent side in a right triangle. Here, the opposite side is 1 foot (rise), and the adjacent side is the given horizontal distance.

Step 3: Calculate the Angle for Case A

For the horizontal distance of 12 ft, use the formula: \( \tan(\theta) = \frac{\text{rise}}{\text{run}} = \frac{1}{12} \). Thus, \( \theta = \tan^{-1}\left(\frac{1}{12}\right) \), which gives approximately \( \theta \approx 4.76^\circ \).

Step 4: Calculate the Angle for Case B

For the horizontal distance of 8 ft, use the formula: \( \tan(\theta) = \frac{1}{8} \). Thus, \( \theta = \tan^{-1}\left(\frac{1}{8}\right) \), which gives approximately \( \theta \approx 7.13^\circ \).

Step 5: Calculate the Angle for Case C

For the horizontal distance of 20 ft, use the formula: \( \tan(\theta) = \frac{1}{20} \). Thus, \( \theta = \tan^{-1}\left(\frac{1}{20}\right) \), which gives approximately \( \theta \approx 2.86^\circ \).

Key Concepts

Right Triangle

Understanding a right triangle is crucial when working with ramps and angles. A right triangle is a three-sided polygon with one angle measuring exactly 90 degrees. In the context of our ramp problem, the right triangle is formed by:

• The vertical rise of the ramp (1 foot) which acts as one of the shorter sides.
• The horizontal run, which is the length of the base of the triangle.
• The hypotenuse, which in this case is the ramp itself.

Right triangles are foundational in trigonometry, where one side is perpendicular to another. When faced with scenarios involving ramps, this geometric shape helps us apply specific trigonometric functions to solve problems related to angles and distances.
In practical terms, identifying the rise, run, and hypotenuse of the right triangle allows us to use known formulas to determine unknown angles, such as the incline of a ramp.

Tangent Function

The tangent function is a key trigonometric function for dealing with angles in a right triangle. Defined as the ratio of the opposite side to the adjacent side, it allows us to find unknown angle measures when two sides of a triangle are known. The formula is expressed as:\[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]In our wheelchair ramp problem, the tangent function helps us find the angle of inclination when the rise (opposite side) and the run (adjacent side) are known.
This function is particularly useful in a variety of real-world applications, such as architecture, physics, and engineering, where precise angle calculations are necessary.

Angle Calculation

Calculating the angle formed by a ramp and the ground is an important step in design and safety. The angle is critical for structural integrity and accessibility requirements. To find this angle using the tangent function, you simply divide the rise by the run and then take the inverse tangent.The steps in determining the angle \( \theta \) for a ramp are:

• Measure the rise (vertical height of the ramp, which is constant at 1 foot in this exercise).
• Measure the run (horizontal distance of the ramp's base).
• Apply the tangent function formula.

With the correct calculations, the resulting angle ensures that the ramp meets legal and practical standards, such as those for wheelchair accessibility.

Inverse Tangent

The inverse tangent, or \( \tan^{-1} \), is a mathematical function used to find an angle when the tangent value is known. This operation is essential in solving problems where the tangent of an angle is established, and the goal is to determine the actual angle measurement. In mathematical notation, if \( \tan(\theta) = x \), then \( \theta = \tan^{-1}(x) \).For the wheelchair ramp scenario:

• Determine the tangent of the angle using the ratio of rise to run.
• Apply the inverse tangent function to calculate the angle.

Utilizing the inverse tangent in ramp design helps evaluate angles against requirements, aiding smooth and safe transition for users.

Ramp Design

Designing a ramp, especially for wheelchair accessibility, involves careful calculations to ensure optimal use and safety. The angle of a ramp is influenced by the rise and horizontal run, affecting its steepness. Several key considerations in ramp design include:

• Legal requirements: These often dictate the maximum allowable slope for ramps to ensure safety and accessibility.
• Environmental factors: Such as anticipating snow accumulation, which can affect the recommended slope.
• Usage needs: Different locations and applications may necessitate unique ramp designs.

By calculating the correct angle of inclination, ramp designs can meet both safety standards and functional needs, providing easy and practical access for all users.