Question

A road that rises \(1 \mathrm{ft}\) for every \(100 \mathrm{ft}\) traveled horizontally is said to have a 1\% grade. Portions of the Lewiston grade, near Lewiston, Idaho, have a \(6 \%\) grade. At what angle is this road inclined above the horizontal?

Short answer

The road is inclined at approximately 3.43 degrees above the horizontal.

Step-by-step solution

Understand the Problem

We need to find the angle of inclination above the horizontal for a road with a 6% grade. A 6% grade means the road rises 6 feet for every 100 feet of horizontal distance.

Identify the Mathematical Relationship

The grade of the road can be seen as the tangent of the angle of inclination. Therefore, for a grade of 6%, we have the equation \( \tan(\theta) = \frac{6}{100} = 0.06 \).

Calculate the Angle Using the Inverse Tangent

To find the angle \( \theta \), we take the inverse tangent (arctan) of 0.06. Thus, \( \theta = \arctan(0.06) \).

Compute the Angle

Using a calculator, compute \( \theta \). \( \arctan(0.06) \approx 3.43^{\circ} \). So, the inclination angle above the horizontal is approximately 3.43 degrees.

Key concepts

Grade of a Road

The grade of a road is a way to describe its steepness. When engineers design a road, they often refer to its "grade" to convey how much the road rises over a certain horizontal distance. A grade is usually expressed as a percentage. This percentage represents the rise over a 100 feet horizontal run. For example:

• A 1% grade means the road rises 1 foot for every 100 feet horizontally.
• A 6% grade means a rise of 6 feet for every 100.

This concept is crucial for understanding and designing roadways, especially in hilly or mountainous areas.

Inverse Tangent

The inverse tangent, also known as arctan, is a function used in trigonometry to find angles. If you have a certain tangent value and you want to know the angle whose tangent is that value, you would use the inverse tangent function. In mathematical terms, if \( \tan(\theta) = x \), then \( \theta = \arctan(x) \). The purpose of this function is to find the angle when the ratio of the opposite side to the adjacent side of a right triangle is known. This is specifically useful for determining the inclination angle for different grades.

Tangent Function

The tangent function is one of the basic trigonometric functions, alongside sine and cosine. In a right triangle, the tangent of an angle is defined as the ratio of the side opposite to the angle to the side adjacent to it. Mathematically, it is expressed as: \[\tan(\theta) = \frac{\text{Opposite side}}{\text{Adjacent side}} \]Understanding the tangent function is essential because it's directly related to the concept of slope and grade, making it a vital tool in various practical and scientific applications. For road grades, the tangent function simplifies the relationship between rise and run, aiding in angle determination for slopes.

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between angles and sides of triangles. It is fundamentally about studying right triangles, especially using functions like sine, cosine, and tangent, which define these relationships. Understanding trigonometry allows us to solve problems involving triangles, waves, and oscillations.

• It helps in calculating distances and angles in navigation and astronomy.
• In the context of roads, it lets us find the angle of inclination using grades (as demonstrated in the solution of the exercise).

Thus, trigonometry becomes an invaluable tool for civil engineering, architecture, physics, and more.

Mathematical Relationship

When discussing grades and angles, mathematical relationships tell us how these quantities are interconnected. For slopes and grades, we explore how they define angles through trigonometric functions. Consider:

• The tangent function expresses the relationship between the vertical rise and horizontal distance.
• The inverse tangent helps in finding the angle from this known relationship.

In simple terms, the mathematical relationship between a grade and an angle is expressed by the tangent function, where finding the angle requires applying the inverse tangent to the grade as a ratio. This link is crucial in solving real-world problems where inclination and slope are factors.