Problem 257 - Precalculus

Question

The grade of a road is 7\(\%\) . This means that for every horizontal distance of 100 feet on the vertical rise is 7 feet. Find the angle the road makes with the horizontal in radians.

Step-by-Step Solution

Answer

The angle with the horizontal is approximately 0.0698 radians.

Step 1: Understanding the Problem

We are given that the grade of the road is 7\(\%\), which means that the vertical rise (opposite side) over a horizontal distance (adjacent side) of 100 feet is 7 feet. We need to find the angle the road makes with the horizontal in radians.

Step 2: Setting Up the Trigonometric Ratio

The grade of the road can be interpreted as a tangent of the angle \(\theta\) because tangent is the ratio of the opposite side to the adjacent side in a right triangle. Therefore, \( \tan \theta = \frac{7}{100} \).

Step 3: Calculating the Angle in Radians

To find the angle \(\theta\), take the arctangent (inverse tangent) of \( \frac{7}{100} \). Thus, \( \theta = \arctan\left(\frac{7}{100}\right) \). Use a calculator to find this value in radians.

Step 4: Converting the Angle from Degrees to Radians (optional)

If the calculator provides the angle in degrees, convert it to radians by multiplying by \(\frac{\pi}{180}\). However, most scientific calculators can directly provide the angle in radians upon request.

Step 5: Final Answer

The angle \(\theta\) turned out to be approximately \(0.0698\) radians based on the calculations.

Key Concepts

Grade of a Road

When we talk about the 'grade of a road,' we are referring to its steepness or incline. The grade is usually expressed as a percentage that describes how much the road rises over a certain horizontal distance. For instance, a 7\(\%\) grade means that for every 100 horizontal feet, the road rises 7 feet vertically.

A road's grade is essential for vehicle safety and performance. A higher grade indicates a steeper incline, which can affect fuel efficiency and brake performance. Understanding the grade is crucial for road design, ensuring the incline is navigable for both vehicles and pedestrians.

• Calculation: Grade \(\% = \left( \frac{vertical\ rise}{horizontal\ distance} \right) \times 100\% \)
• 7 feet rise per 100 feet horizontal means the road has a 7\% grade.
• This corresponds to a rise of 0.07 per unit of horizontal distance.

Tangent Ratio

The tangent ratio is a fundamental concept in trigonometry and is particularly useful in problems involving right triangles.

In a right triangle, the tangent of an angle \( \theta \) is the ratio of the length of the opposite side to the length of the adjacent side. This ratio is expressed as:

• Tangent formula: \( \tan(\theta) = \frac{opposite}{adjacent} \)

For our road problem, the tangent of the angle is equal to the grade of the road divided by the horizontal distance: \( \tan(\theta) = \frac{7}{100}\).

The tangent functions are extremely useful in real-world scenarios, like calculating slopes and angles in engineering and architecture.

Angle in Radians

In trigonometry, angles can be measured in degrees or radians. Radians are a way of measuring angles based on the radius of a circle. A circle has \(2\pi\) radians, which is equivalent to 360 degrees.

Radians offer a more natural way of describing angles, especially in calculus and advanced mathematics. For example, one radian is the angle created when the arc length equals the radius of the circle.

• 1 radian \( \approx 57.2958 \) degrees.

In the problem, when calculating the angle using the inverse tangent, we find \( \theta \approx 0.0698 \) radians.

Understanding radians is key to working effectively with various trigonometric functions and calculus problems.

Inverse Tangent

The inverse tangent function, often denoted as \( \arctan \) or \( \tan^{-1} \), is used to find an angle when you know the tangent value.

In our example, we know the tangent of the angle due to the road's grade relationship \( \tan(\theta) = \frac{7}{100} \). To find the angle \( \theta \), we use the inverse tangent function:

• \( \theta = \arctan\left(\frac{7}{100}\right) \)

This mathematical function effectively "reverses" the tangent process, finding the angle that gives a specific tangent ratio. Most calculators have an \( \arctan \) button or function, which allows you to quickly compute this value.

While calculators can provide results in degrees or radians, using radians is often preferable in advanced mathematics. This is because radians align more intuitively with other mathematical concepts, making functions like the inverse tangent easier to interpret in broader contexts.