Problem 9 - Algebra 2 and Trigonometry

Question

A wheelchair user brings along a 6 -foot long portable ramp to get into a van. For safety and ease of whecling, the ramp should make a 5 - to 10 -degree angle with the ground. a. Let \(\theta\) represent the measure of the angle that the ramp makes with the ground in radians. Express, as a function of \(\theta\) , the distance \(d\) between the foot of ramp and the base of the van on which the ramp sits. b. What is the domain of the function from part a? c. Graph the function from part a using the domain found in part b. d. What is the smallest safe distance from the foot of the ramp to the base of the van?

Step-by-Step Solution

Answer

The smallest safe distance, \( d \), is approximately 5.91 feet when \( \theta = 10^\circ \). The function is \( d(\theta) = 6 \cos(\theta) \) with domain \( [\frac{\pi}{36}, \frac{\pi}{18}] \).

Step 1: Understand the problem context

We are given a 6-foot ramp used by a wheelchair to access a van. The ramp forms angle \(\theta\) with the ground, and this angle must be between 5 and 10 degrees for safety. We need to calculate the horizontal distance \(d\) based on this angle.

Step 2: Express distance d as a function of θ

The distance \(d\) between the foot of the ramp and the base of the van relates to the hypotenuse (the ramp) via the cosine function, where \(\cos(\theta) = \frac{d}{6}\). Therefore, we have the function: \(d(\theta) = 6 \cos(\theta)\).

Step 3: Determine the domain of the function

The angle \(\theta\) should be between 5 and 10 degrees, but we need to convert these to radians. Since \(1^\circ = \frac{\pi}{180}\) radians, \(5^\circ = \frac{\pi}{36}\) and \(10^\circ = \frac{\pi}{18}\). So, the domain is \(\frac{\pi}{36} \leq \theta \leq \frac{\pi}{18}\).

Step 4: Sketch the graph of the function

Plot the function \(d(\theta) = 6 \cos(\theta)\) over the interval \(\theta \in [\frac{\pi}{36}, \frac{\pi}{18}]\). The graph will show the relationship between the angle \(\theta\) and the distance \(d\).

Step 5: Determine the smallest distance for safety

For the smallest safe distance, calculate \(d\) using the maximum angle \(\theta = \frac{\pi}{18}\) from the domain. Compute \(d(\theta) = 6 \cos\left(\frac{\pi}{18}\right)\) using the cosine of 10 degrees.

Key Concepts

Understanding the Cosine Function

The cosine function is essential in trigonometry and often associated with right-angled triangles. It helps us relate the angle to the lengths of the sides of the triangle.

• In a right-angled triangle, the cosine of an angle \( \theta \) is the ratio of the adjacent side to the hypotenuse.
• This means if you know the hypotenuse length and the angle, you can find the adjacent side length using the formula: \( \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \).

In the context of the ramp problem, the ramp itself serves as the hypotenuse, and the horizontal distance (the distance from the bottom of the ramp to the van) is the adjacent side.
Hence, we express this relationship as \( d(\theta) = 6 \cos(\theta) \), where 6 is the length of the ramp.
This provides a practical application: ensuring safe wheelchair access by calculating the correct horizontal distance based on the ramp's angle and length.

Exploring Radian Measure

Radian measure is pivotal in calculus and trigonometry, especially when dealing with angles. Instead of degrees, angles are represented with radians, which offer a more direct mathematical relationship.

• One full rotation around a circle is \( 2\pi \) radians, equivalent to 360 degrees. Therefore, \( 1\pi \) radian represents 180 degrees.
• Smaller fractions of \( \pi \) correspond to smaller angles, offering a precise way to define and work with angles.

In our ramp scenario, we need to convert the angle restrictions from degrees into radians: \( 5^\circ \approx \frac{\pi}{36} \) and \( 10^\circ \approx \frac{\pi}{18} \).
This transformation is crucial because it translates the problem into the mathematical language used for graphing and function analysis.
Understanding radians helps in sketching the graph over the correct domain and enables accurate calculations of \( d(\theta) \) for real-world applications.

Graphing Functions

Graphing functions, like \( d(\theta) = 6 \cos(\theta) \), makes it easier to visualize mathematical relationships.

• The x-axis represents the angle \( \theta \), while the y-axis represents the distance \( d \), where each point on the graph corresponds to a particular angle and its resulting distance.
• In our specific example, \( \theta \) varies between \( \frac{\pi}{36} \) and \( \frac{\pi}{18} \).

The graph will typically show how \( d \) (the distance) changes smoothly as the angle becomes larger within the given range.

This visual tool aids in identifying trends or verifying mathematical assumptions. For instance, you can observe that as the angle increases, the horizontal distance decreases.
Graphing the function helps ensure that the ramp will function safely at all the specified angles, giving a visual representation of safety thresholds.