Problem 22 - Essential Calculus

Question

Use a graphing utility to approximate the points of intersection of the graphs of the polar equations. Confirm your results analytically. $$ \begin{aligned} r &=3(1-\cos \theta) \\ r &=\frac{6}{1-\cos \theta} \end{aligned} $$

Step-by-Step Solution

Answer

The points of intersection of the two polar equations are (0,0) and (6, \pi).

Step 1: Graph the functions

Using a graphing utility, plot the two polar functions \( r = 3(1 - cos(\theta)) \) and \( r = \frac{6}{1 - cos(\theta)} \). Observe that the graphs intersect at two points.

Step 2: Set the functions equal to each other

To find the precise points of intersection, set the two functions equal to each other and solve for \( \theta \):\n3(1 - cos(\theta)) = \frac{6}{1 - cos(\theta)}. Cross-multiplication yields: \(3 - 3cos(\theta) = 6 - 6cos(\theta)\). Solving for \( \theta \), we get \( \theta = 0 \) or \( \theta = \pi \).

Step 3: Substitute values

Now, substitute \( \theta = 0 \) or \( \theta = \pi \) into the original equations to get the \( r \) values. \nFor \( \theta = 0 \), both equations yield \( r = 0 \) and for \( \theta = \pi \), both equations yield \( r = 6 \).

Key Concepts

Graphing Utilities

Graphing utilities, like graphing calculators or software, are essential tools in visualizing mathematical concepts, especially complex ones like polar coordinates. They help display graphs which can be tricky to sketch by hand. For polar equations such as

• \( r = 3(1 - \cos \theta) \)
• \( r = \frac{6}{1 - \cos \theta} \)

we use a graphing utility to plot these equations in polar coordinates.
Polar coordinates involve a radius \( r \) and an angle \( \theta \), contrasting with Cartesian coordinates. These graphs form loops and curves unique to their equations, making graphing utilities crucial to avoid errors and save time.
By inputting polar equations into the utility, students can precisely determine how these graphs interact, such as identifying where they intersect. This lays the foundation for solving more complex mathematical problems.

Intersection Points

Finding the intersection points of polar graphs is an important skill that involves determining where two curves meet. In this exercise, we are interested in the intersections of

• \( r = 3(1 - \cos \theta) \)
• \( r = \frac{6}{1 - \cos \theta} \)

Graphing utilities visually show these intersection points, but to ensure accuracy, we need to calculate them analytically.
We start by equating the two equations: \[3(1 - \cos(\theta)) = \frac{6}{1 - \cos(\theta)}\]Solving this equation simplifies the process of finding specific values of \( \theta \) where the intersections occur. Upon solving, we determine \( \theta = 0 \) and \( \theta = \pi \), and can then substitute these values back into the original equations. This way, we confirm these angles provide consistent radial values \( r \) across both equations, verifying the intersection points.

Analytical Confirmation

After approximating the intersection points visually with a graphing utility, it's crucial to perform an analytical confirmation. This step ensures that the conclusions drawn from the graph are mathematically sound. To achieve this, we need to analytically solve the equation \[3(1 - \cos(\theta)) = \frac{6}{1 - \cos(\theta)}\]By cross-multiplying and simplifying, this equation transforms to find \( \theta \). Our resulting solution is \( \theta = 0 \) or \( \theta = \pi \).
Substituting these back into the original polar equations confirms consistent \( r \) values:

• For \( \theta = 0 \), \( r = 0 \)
• For \( \theta = \pi \), \( r = 6 \)

Thus, the points of intersection are confirmed as \( (0,0) \) and \( (6,\pi) \). This verification process is vital because graphical approximations may sometimes lead to human or computational errors, and analytical methods act as a solid check to support our findings.