Problem 72 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. A wheelchair ramp must be constructed so that the slope is not more than 1 inch of rise for every 1 foot of run, so I used the tangent function to determine the maximum angle that the ramp can make with the ground.

Step-by-Step Solution

Answer

Yes, the statement makes sense since the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Therefore, using the tangent function to calculate the maximum angle that the ramp can make with the ground when given the ratio of rise to run is appropriate. The angle can be calculated as \( \theta = \tan^{-1} (\frac{1}{12}) \).

Step 1: Understanding the statement

The statement talks about a wheelchair ramp that should not have a slope of more than 1 inch of rise for every 1 foot of run. This is a requirement for the construction of wheelchair ramps for them to be safe and accessible. The individual intends to use the tangent function to determine the maximum angle that the ramp can make with the ground.

Step 2: Analyzing the use of tangent

The use of tangent here makes sense because the tangent of an angle in a right triangle is defined as the ratio of the side opposite to the angle (rise) to the side adjacent to the angle (run). So, if we let the angle the ramp makes with the ground be \( \theta \), then \( \tan(\theta) = \frac{rise}{run} \) and we are given that \( \frac{rise}{run} = \frac{1 inch}{1 foot} = \frac{1}{12} \), because 1 foot equals 12 inches.

Step 3: Calculating the angle

To find the angle \( \theta \), which makes with the ground we can use the arctangent function or inverse tangent function. If we plug our ratio into the inverse tangent function, we get \( \theta = \tan^{-1} (\frac{1}{12}) \). This calculates the maximum angle the ramp can make with the ground in degrees.

Key Concepts

Slope of Ramp

When it comes to understanding the practical implications of geometry, the concept of a ramp's slope offers a clear example. In the context of a wheelchair ramp, the slope is crucial as it affects accessibility and safety. The slope represents the incline level of the ramp and is usually described by the rise over the run ratio. In simpler terms, it's the vertical height increase per horizontal distance traveled.

For wheelchair ramps, there is a standard to ensure ease of use and safety: the slope must not exceed 1 inch of rise for every 12 inches of run. This ensures the ramp is not too steep for users. By applying the tangent function, which we'll explore further in the next sections, we can mathematically determine the appropriate angle to achieve this slope, thus integrating accessibility guidelines with trigonometric principles.

Trigonometry in Real Life

Trigonometry, though often perceived as an abstract field of mathematics, has a plethora of applications in everyday life. The calculation of wheelchair ramp angles is just one of the endless examples of trigonometry in real-world situations. Trigonometry is the study of relationships between angles and sides in triangles, particularly right triangles. It's used in various fields such as architecture, engineering, astronomy, and even in the design of video games.

Understanding the practical applications of trigonometry can enhance a student's learning experience as it connects theoretical math with tangible scenarios. Knowing how to use trigonometric functions to solve problems helps in making informed decisions in design, construction, and even navigation, making it a valuable tool in many careers.

Inverse Tangent Function

The inverse tangent function, often denoted as \(\tan^{-1}\) or arctan, is a fundamental component of trigonometry used to find an angle when the tangent value is known. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. The inverse tangent function is the reverse operation; it provides the angle that corresponds to a particular tangent value.

For the ramp problem we have discussed, by plugging the ratio \(\frac{1}{12}\) into the inverse tangent function, we get the value of the angle \(\theta\) in degrees or radians. This angle is the maximum angle the ramp can make with the ground while adhering to the safety standards. It's essential for students to grasp this concept as it shows how an abstract mathematical function can yield practical, everyday solutions.

Right Triangle Angles

Right triangle angles are at the core of trigonometry. A right triangle is distinguished by the presence of one 90-degree angle, and the relationships among the angles and sides of these triangles are consistent and predictable, allowing for the practical application of trigonometric functions. These functions, including sine, cosine, and tangent, are ratios that allow for the calculation of unknown angles and side lengths given certain known values.

Recognizing these relationships enables one to solve a wide array of practical problems. In the case of a wheelchair ramp, understanding that the angle formed with the ground, along with the lengths of the adjacent and opposite sides of the triangle (run and rise, respectively), fits within the framework of a right triangle. This enables the precise calculation of the necessary angle through trigonometric functions, thus bridging the gap between theory and practice.