Problem 48 - Precalculus with Calculus Previews

Question

Use a graphing utility to obtain the graphs of the given three polar equations on the same rectangular coordinate system. Use different colors for each graph. $$ \begin{aligned} &\text { Inner orbit } r=\frac{73.5}{1+0.05 \cos \theta}, \text { Hohmann }\\\ &\text { transfer } r=\frac{77}{1_{-}+0.1 \cos \theta}, \text { outer orbit }\\\ &r=\frac{84.7}{1+0.01 \cos \theta} \end{aligned} $$

Step-by-Step Solution

Answer

Use a graphing tool to plot and distinguish the three orbits by color.

Step 1: Understand the Polar Equation

Each given polar equation is in the form \( r = \frac{k}{1 + e \cos \theta} \), which represents a conic section. These specific forms are for orbits in polar coordinates, where \( k \) is the semi-latus rectum and \( e \) is the eccentricity. The equations are:1. Inner orbit: \( r = \frac{73.5}{1 + 0.05 \cos \theta} \)2. Hohmann transfer: \( r = \frac{77}{1 - 0.1 \cos \theta} \)3. Outer orbit: \( r = \frac{84.7}{1 + 0.01 \cos \theta} \).

Step 2: Set Up the Graphing Utility

Prepare the graphing utility, such as a graphing calculator or software, and ensure it's capable of plotting polar equations. You will plot three separate equations on the same screen.

Step 3: Plot the Inner Orbit Graph

Enter the equation \( r = \frac{73.5}{1 + 0.05 \cos \theta} \) into the graphing utility. Choose a color (e.g., blue) to represent this orbit. Plot the graph within the range of \( \theta \) from 0 to \( 2\pi \) to observe the complete orbit.

Step 4: Plot the Hohmann Transfer Graph

Enter the equation \( r = \frac{77}{1 - 0.1 \cos \theta} \) into the graphing utility. Use a different color (e.g., red) to distinguish this transfer orbit from the others. Again, plot \( \theta \) from 0 to \( 2\pi \).

Step 5: Plot the Outer Orbit Graph

Enter the equation \( r = \frac{84.7}{1 + 0.01 \cos \theta} \) into the graphing utility. Assign a unique color (e.g., green) for this orbit. Plot \( \theta \) from 0 to \( 2\pi \) for complete visualization.

Step 6: Analyze and Compare the Graphs

Once all graphs are plotted on the same coordinate system, analyze how the orbits differ. Observe the size and shape based on varying eccentricities and semi-latus rectums. These differences reflect the orbits' characteristics.

Key Concepts

Graphing Utility

A graphing utility is a powerful tool used to visualize mathematical equations, especially in scenarios where plotting by hand would be complex or time-consuming. In the context of polar equations, a graphing utility can help you plot functions like \( r = \frac{k}{1 + e \cos \theta} \), which are not linear and can demonstrate interesting patterns and shapes that change with different values of \( k \) and \( e \). This is incredibly useful when dealing with multiple equations, allowing for color differentiation between graphs.
To effectively utilize a graphing utility:

• Ensure your software or calculator is capable of plotting polar equations. Most modern graphing utilities have this feature.
• Input each equation separately, selecting different colors to distinguish between them. This visual differentiation helps in comparing the orbits.
• Set the range of \( \theta \) typically from 0 to \( 2\pi \) to view one complete cycle of the equation's graph.
• Adjust the viewing window to ensure all parts of the graph are visible, helping you analyze positions and orientations effectively.

Using a graphing utility can deepen your understanding of how equations behave, making it easier to see connections between variables and their graphical representations.

Conic Sections

Conic sections are the curves obtained by the intersection of a cone with a plane. In the context of polar equations, conic sections like circles, ellipses, parabolas, and hyperbolas can be represented. For polar coordinates, the general form \( r = \frac{k}{1 + e \cos \theta} \) hints at these conic shapes, with \( k \) and \( e \) determining their specific type and orientation.
Understanding conic sections involves recognizing how different conics relate to orbits:

• If the eccentricity \( e = 0 \), the orbit is a circle.
• For \( 0 < e < 1 \), the orbit is an ellipse, the most common shape representing orbits in this format.
• When \( e = 1 \), you have a parabola, which isn't typical for orbital paths.
• For \( e > 1 \), the orbit would form a hyperbola.

In our exercise, the given polar equations represent variations of ellipses, illustrating how they model unique orbital paths when classified by eccentricity. Recognizing these shapes gives you insight into how celestial orbits function, such as those of planets and satellites.

Eccentricity

Eccentricity is a key parameter in defining the shape of a conic section. It is denoted by the symbol \( e \), and measures how much a conic section deviates from being circular. In our polar equations, eccentricity influences how stretched or elongated an orbit appears.
The eccentricity values from the exercise are:

• Inner orbit's eccentricity: \( e = 0.05 \)
• Hohmann transfer's eccentricity: \( e = 0.1 \)
• Outer orbit's eccentricity: \( e = 0.01 \)

With knowing \( e \):

• A lower eccentricity means the orbit is closer to a perfect circle, as seen in the Outer orbit.
• Higher eccentricity values like in the Hohmann transfer indicate more elongated ellipses.

Understanding eccentricity helps predict orbital dynamics. It clarifies the motion path of planets, satellites, and space-bound trajectories, helping us visualize the variety of motions in our cosmos.