Problem 16 - Thomas Calculus

Question

Water main construction \(A\) water main is to be constructed with a 20\(\%\) grade in the north direction and a 10\(\%\) grade in the east direction. Determine the angle \(\theta\) required in the water main for the turn from north to east.

Step-by-Step Solution

Answer

The angle required for the water main turn is approximately 5.57°.

Step 1: Understanding the Problem

We need to find the angle \( \theta \) formed between a line with a northward grade of 20% and another with an eastward grade of 10%. The grades can be interpreted as slopes of the lines.

Step 2: Defining Slopes

A 20% grade means the northward slope is \( m_1 = 0.2 \) and a 10% grade eastward means the eastward slope is \( m_2 = 0.1 \). We consider both slopes for our calculation.

Step 3: Using the Angle Between Lines Formula

The formula to find the angle \( \theta \) between two lines with slopes \( m_1 \) and \( m_2 \) is:\[\tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|\]

Step 4: Substitute Values into the Formula

Substituting \( m_1 = 0.2 \) and \( m_2 = 0.1 \) into the formula:\[\tan(\theta) = \left| \frac{0.1 - 0.2}{1 + 0.2 \cdot 0.1} \right| = \left| \frac{-0.1}{1.02} \right| = \frac{0.1}{1.02}\]

Step 5: Calculating the Angle

Calculate the angle \( \theta \) using the arctangent function:\[\theta = \arctan\left(\frac{0.1}{1.02}\right)\]Using a calculator:\[\theta \approx 5.57^\circ\]

Step 6: Conclusion

The angle \( \theta \) required for the water main to turn from north to east is approximately \( 5.57^\circ \).

Key Concepts

Angle between lines

The angle between lines is an important concept in geometry and calculus. It involves determining the angle formed by two intersecting lines, each having its slope.
Finding this angle is useful in numerous applications, such as engineering, physics, and computer graphics. To calculate this angle, we often use a standard formula that connects the slopes of the lines to the tangent function.
For two lines with slopes \(m_1\) and \(m_2\), the formula to find the angle \(\theta\) between them is:

• \( \tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \)

This formula considers the differences in slopes and whether the lines are parallel, perpendicular, or intersecting at another angle. Remember that the slopes determine the steepness or inclination of the lines.
When you substitute the values into this formula, you can find the value of \(\theta\). However, if \(m_1 m_2 = -1\), the lines are perpendicular, and \(\theta\) is 90 degrees. For parallel lines, the slopes are equal, making the angle between them zero degrees.

Tangent function

The tangent function is fundamental in trigonometry. It relates the angles and slopes, especially useful when finding angles between lines.
Mathematically, the tangent of an angle \(\theta\) in a right triangle is expressed as the ratio of the opposite side to the adjacent side. In formula form:

• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

In the context of calculus and slopes, the tangent function helps find an angle \(\theta\) between two lines that have known slopes. The function involves calculating differences and products of slopes.
When the slopes are plugged into the formula \(\tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|\), we can then use the inverse tangent or arctangent function \(\arctan\) to find the actual angle. The arctangent function returns the angle whose tangent is a given number.

Slope calculation

The slope is a measure of the steepness or the inclination of a line. In terms of calculus, it is the rate of change of the function - essentially, how one variable changes in responses to another.
For straight lines, the slope \(m\) is calculated by the formula:

• \( m = \frac{\Delta y}{\Delta x} \)

where \(\Delta y\) is the change in the y-coordinate, and \(\Delta x\) is the change in the x-coordinate. Thus, it expresses the vertical change per unit of horizontal change.
When it comes to grades, like those given as percentages in construction or navigation, the slope can be interpreted as a grade percentage. For instance, a 20% grade means a vertical change of 0.2 units for each 1 unit change horizontally. This concept is foundational in determining angles between lines and understanding their orientation in space.