Problem 45 - Precalculus with Limits

Question

In Exercises 37-46, find the angle \(\theta\) (in radians and degrees)between the lines. \(0.05x - 0.03y = 0.21\) \(0.07x + 0.02y = 0.16\)

Step-by-Step Solution

Answer

The angle between the lines is approximately 0.8392 radians or 48.09 degrees.

Step 1: Find the Slopes of the Lines

The equations are in the form \(Ax+By=C\). To find the slope, rearrange the equations into slope-intercept form \(y=mx+b\), where \(m\) is the slope. The first line becomes \(y = \frac{5}{3}x - 7\) and the second line becomes \(y = -\frac{7}{2}x + 8\). Therefore, the slopes are \(m_1 = \frac{5}{3}\) for the first line and \(m_2 = -\frac{7}{2}\) for the second line.

Step 2: Find the Angle Between the Lines in Radians

The formula for the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is \(\theta = \arctan\left|\frac{m_2 - m_1}{1 + m_1m_2}\right|\). Plugging in the lengths gives \(\theta = \arctan\left|\frac{-\frac{7}{2} - \frac{5}{3}}{1 + \left(\frac{5}{3}\right)\left(-\frac{7}{2}\right)}\right| = \arctan\left|\frac{-29/6}{1 - \frac{35}{6}}\right| = \arctan\left|-1.161\right| = 0.8392 \) radians.

Step 3: Convert Radians to Degrees

To convert the angle from radians to degrees, use the formula \(degrees = radians \times \frac{180}{\pi}\). Plugging in gives \(\theta = 0.8392 \times \frac{180}{\pi} = 48.09\) degrees.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express the equation of a straight line. It is typically written as \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) represents the y-intercept, which is the point where the line crosses the y-axis.

To convert an equation from standard form \(Ax + By = C\) to slope-intercept form, we isolate \(y\) on one side. For example, let's consider some line with the equation \(0.05x - 0.03y = 0.21\). To put this into slope-intercept form, you would do the following steps:

• Add \(0.03y\) to both sides to get \(0.05x = 0.03y + 0.21\).
• Next, you would divide every term by \(0.03\) to solve for \(y\), resulting in \(y = \frac{5}{3}x - 7\).

This gives us the slope, \(m = \frac{5}{3}\), and y-intercept, \(b = -7\). Understanding how to find the slope and y-intercept from an equation is crucial for graphing lines and analyzing their relationship to each other, such as when finding angles between lines.

Angle Between Lines

To find the angle between two lines, we look at their slopes. The angle \(\theta\) between the lines can be found using the formula: \[ \theta = \arctan\left|\frac{m_2 - m_1}{1 + m_1m_2}\right| \], where \(m_1\) and \(m_2\) are the slopes of the two lines. If the lines are perpendicular, the angle between them is \(\frac{\pi}{2}\) radians or 90 degrees; if they are parallel, the angle is 0.

Let's apply this to our example where line one has a slope \(m_1 = \frac{5}{3}\) and line two has a slope \(m_2 = -\frac{7}{2}\). By plugging these slopes into the formula, we calculate the angle between the lines in radians. The absolute value ensures the angle is always positive, as the direction (clockwise or counterclockwise) of the angle is not considered here. This formula essentially measures the smallest angle between the two lines, providing an essential concept for geometry and trigonometry related problems.

Radians to Degrees Conversion

Radians and degrees are both units used to measure angles. The full circle in degrees is 360°, while in radians is \(2\pi\) radians. To convert an angle from radians to degrees, the formula is: \[degrees = radians \times \frac{180}{\pi}\].

Applying this to our calculated angle \(\theta = 0.8392\) radians, we get: \(0.8392 \times \frac{180}{\pi} = 48.09\) degrees. This is a crucial step in problem-solving as many real-world applications and further mathematics use degrees, so understanding how to transition between these two units allows for a more comprehensive understanding of angles and their measurements in various contexts.