Problem 5 - The Calculus with Analytic Geometry

Question

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{l}r=4 \theta \\ r=\frac{1}{2} \pi\end{array}\right.\)

Step-by-Step Solution

Answer

The intersection point is \( \left( \frac{\pi}{2}, \frac{\pi}{8} \right) \).

Step 1: Understand the Equations

There are two polar equations given: 1) \( r = 4\theta \) 2) \( r = \frac{1}{2}\pi \)

Step 2: Set Equations Equal to Each Other

To find the points of intersection, set the two equations equal to each other: \( r = 4\theta = \frac{1}{2}\pi \)

Step 3: Solve for \(\theta\)

Solve the equation \( 4\theta = \frac{1}{2}\pi \) for \(\theta\): \[ 4\theta = \frac{1}{2}\pi \ \theta = \frac{\pi}{8} \]

Step 4: Find the Corresponding \(r\)

Substitute \( \theta = \frac{\pi}{8} \) back into one of the original equations to find \( r \): \[ r = 4\theta \ r = 4 \left( \frac{\pi}{8} \right) = \frac{4\pi}{8} = \frac{\pi}{2} \]

Step 5: Determine Intersection Points

The intersection point in polar coordinates is: \[ \left( r, \theta \right) = \left( \frac{\pi}{2}, \frac{\pi}{8} \right) \]

Step 6: Sketch the Graphs

Draw the graphs for both equations on the same polar axis: - For \( r = 4\theta \), the graph is a spiral starting from the origin.- For \( r = \frac{\pi}{2} \), the graph is a circle with a radius of \( \frac{\pi}{2} \). The intersection point \( \left( \frac{\pi}{2}, \frac{\pi}{8} \right) \) should be marked on the graph.

Key Concepts

Polar Equations

Polar equations describe relationships between the radius, r, and the angle, \( \theta \), in polar coordinates. Unlike Cartesian coordinates that use (x, y) positions, polar coordinates use (r, \( \theta \)) to map points.
In the given exercise, we have two polar equations:

• \( r = 4\theta \)
• \( r = \frac{1}{2}\pi \)

Both equations define curves in the polar coordinate system. The first one, \( r = 4\theta \), represents a spiral that increases in radius as the angle \( \theta \) increases. The second, \( r = \frac{1}{2}\pi \), represents a circle with a constant radius of \( \frac{\pi}{2} \). Understanding these basic forms can help us visualize their intersections.

Graph Sketching

Sketching the graphs of polar equations is a crucial step in understanding their behavior and points of intersection. For the given equations:
- **Spiral Equation \( r = 4\theta \)**: This graph starts at the origin (0, 0) and winds outward as \( \theta \) increases. Each time \( \theta \) increases, the radius, r, also increases proportionally.
- **Circle Equation \( r = \frac{1}{2}\pi \)**: This graph represents a circle centered at the origin with a constant radius \( \frac{\pi}{2} \). No matter the value of \( \theta \), the radius remains the same.
When sketching these graphs on a polar coordinate system, it's important to mark key points and consider the nature of each equation. For instance, the spiral expands continuously while the circle maintains a fixed distance from the origin. Drawing these curves accurately will highlight their intersection points more clearly.

Solving for Intersection Points

Intersection points are the coordinates where two polar graphs meet. To find these points for the given equations, follow these steps:
1. **Set the Equations Equal to Each Other**: To determine where the two graphs intersect, equate the two polar equations:
\(4\theta = \frac{1}{2}\pi \).
2. **Solve for \( \theta \)**: Isolate \( \theta \) in the equation:
\[ 4\theta = \frac{1}{2}\pi \]
\(\theta = \frac{\pi}{8}\)
3. **Find Corresponding r**: Substitute \( \theta = \frac{\pi}{8} \) back into one of the original equations to find the radius, r.
Using \( r = 4\theta \):
\[ r = 4 \left( \frac{\pi}{8} \right) = \frac{4\pi}{8} = \frac{\pi}{2} \]
Therefore, the intersection point in polar coordinates is \( \left( \frac{\pi}{2}, \frac{\pi}{8} \right) \).
By following these steps methodically, you can solve for intersection points in any pair of polar equations.