Problem 90 - A Graphical Approach to Precalculus with Limits

Question

The term grade has several different meanings in construction work. Some engineers use the term to represent \(\frac{1}{100}\) of a right angle and express it as a percent. For instance, an angle of \(0.9^{\circ}\) would be referred to as a \(1 \%\) grade. (Source: Hay, W., Railroad Engineering. John Wiley and Sons.) (a) By what number should you multiply a grade to convert it to radians? (b) In a rapid-transit rail system, the maximum grade allowed between two stations is \(3.5 \% .\) Express this angle in degrees and in radians.

Step-by-Step Solution

Answer

(a) Multiply by \(\frac{\pi}{200}\), (b) 3.15° or 0.05495 radians.

Step 1: Understanding the grade system

A grade is defined as \(\frac{1}{100}\) of a right angle, and expressed as a percent. Since a right angle is \(90^{\circ}\), a grade of \(1\%\) corresponds to \(0.9^{\circ}\).

Step 2: Convert grade to degrees

We know that \(1\%\) grade is equivalent to \(0.9^{\circ}\), thus a grade expressed as \(g\%\) can be converted to degrees by multiplying it by \(0.9\) since \(\text{degrees} = g \times 0.9\).

Step 3: Convert degrees to radians

First, recall the conversion formula from degrees to radians: \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\). To find the conversion factor directly for grades to radians, use \(\text{radians} = g \times 0.9 \times \frac{\pi}{180}\).

Step 4: Determine the multiplier for grade to radians

Combining steps above, the conversion from grade directly to radians is given by \(\text{radians} = g \times \frac{0.9\pi}{180}\ = g \times \frac{\pi}{200}\). Thus, multiply the grade by \(\frac{\pi}{200}\) to convert to radians.

Step 5: Convert grade to degrees for the rail system

The maximum grade allowed in the rail system is \(3.5\%\). Converted to degrees, this is \(3.5 \times 0.9 = 3.15^{\circ}\).

Step 6: Convert grade to radians for the rail system

Using the convert factor from step 4, convert \(3.5\%\) grade to radians using \(3.5 \times \frac{\pi}{200}\). Calculate the result as \(0.05495 \) radians approximately.

Key Concepts

Grade Percent Conversion

The concept of grade percent conversion is essential in various engineering fields. In construction work, the grade of an angle is defined as a percentage, with each percent representing one-hundredth of a right angle. Since a right angle is 90 degrees, a 1% grade equates to a 0.9-degree angle.

To convert a grade, given as a percentage, into degrees, you simply multiply the grade by 0.9. This conversion is directly based on the relationship between a 1% grade and its equivalent angle. For example, a 3.5% grade can be easily converted to degrees by calculating 3.5 multiplied by 0.9, resulting in 3.15 degrees.

Similarly, to transform grade percentages into radians, a multiplication by \(\frac{\pi}{200}\) is used. This factor stems from integrating the degrees-to-radians conversion \(\left(\frac{\pi}{180}\right)\) into the initial grade-to-degrees conversion formula. This provides a straightforward method to address various mathematical and engineering problems involving angles.

Angle Measurement

Understanding angle measurement is crucial in fields like mathematics and physics. Angles are a measure of rotation and can be expressed in multiple units including degrees and radians.

Degrees are perhaps the most familiar unit and are based on dividing a full circle into 360 equal parts. This means that one degree corresponds to \(\frac{1}{360}\)th of a circle. Thus, common angles such as 90 (a right angle) and 180 (a straight angle) are easily represented.

Radians, on the other hand, offer a more natural unit derived from the properties of a circle. Here, angles are measured in terms of radius lengths, making it highly useful for mathematical calculations involving trigonometric functions. One full circle equals \(2\pi\) radians, so a right angle is equivalent to \(\frac{\pi}{2}\) radians.

Conversions between degrees and radians can be made using the formulas:

• From degrees to radians: \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\)
• From radians to degrees: \(\text{degrees} = \text{radians} \times \frac{180}{\pi}\)

Trigonometry Concepts

Trigonometry, a branch of mathematics, deals with the relationships between side lengths and angles in triangles. These concepts extend beyond simple triangles to apply to cyclical patterns, waves, and oscillatory motion. One of the most fundamental elements is the use of trigonometric functions: sine, cosine, and tangent.

Each of these functions provides a ratio specific to an angle within a right-triangle context:

• Sine: ratio of the length of the opposite side to the hypotenuse.
• Cosine: ratio of the length of the adjacent side to the hypotenuse.
• Tangent: ratio of the length of the opposite side to the adjacent side.

These functions are also essential in converting angles between degrees and radians, as they appear in various mathematical equations, notably those involving waves or rotations. Understanding these trigonometric concepts is essential for solving more complex engineering and physics problems where angles and cycles play a significant role.