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Magazine Chapter 11: Problem 34
Polar coordinates are just the thing for defining spirals. Graph the following spirals. a. \(r=\theta\) b. \(r=-\theta\) c. \(A\) logarithmic spiral: \(r=e^{\theta / 10}\) d. \(A\) hyperbolic spiral: \(r=8 / \theta\) e. An equilateral hyperbola: \(r=\pm 10 / \sqrt{\theta}\) (Use different colors for the two branches.)
Short Answer
Expert verified
Graph polar spirals using given equations, labeling and coloring per description.
Step by step solution
01
Understanding Polar Coordinates
In polar coordinates, a point on the plane is determined by a distance from the origin \(r\) and an angle \(\theta\) from the positive x-axis. The equations provided describe relationships between \(r\) and \(\theta\) to form different spiral curves.
02
Plotting the Spiral \(r=\theta\)
For \(r=\theta\), as \(\theta\) increases, \(r\) also increases linearly. This creates an Archimedean spiral which moves outward from the origin counter-clockwise. To graph this, calculate points for various \(\theta\) values such as 0, \(\pi/2\), \(\pi\), \(3\pi/2\), etc., and plot \((r, \theta)\) in polar coordinates.
03
Plotting the Spiral \(r=-\theta\)
In the equation \(r=-\theta\), as \(\theta\) increases positively, \(r\) decreases negatively, producing an outward spiral in a clockwise direction. The method of plotting points is similar: compute values for \(\theta\) and graph points using polar coordinates.
04
Plotting the Logarithmic Spiral \(r=e^{\theta / 10}\)
This equation represents a logarithmic spiral where \(r\) increases exponentially with \(\theta\). As \(\theta\) grows, \(r\) increases at an increasing rate. Plot values for small to moderate values of \(\theta\) to see the curve expanding outward, recognizing the spiral tightens as it expands.
05
Plotting the Hyperbolic Spiral \(r=8/\theta\)
Here, \(r\) decreases as \(\theta\) increases, creating an inward spiral. To graph, choose \(\theta\) values starting from a small positive number to avoid division by zero and plot the corresponding \(r\) values, noting how the spiral approaches the origin.
06
Plotting the Equilateral Hyperbola \(r=\pm 10/\sqrt{\theta}\)
This represents two branches of a hyperbola. For the positive branch, \(r=10/\sqrt{\theta}\) decreases as \(\theta\) increases, moving outward. For the negative branch, \(r=-10/\sqrt{\theta}\), plot where \(r\) is negative, moving inward. Use different colors for clarity when plotting both branches, emphasizing symmetry about the origin.
07
Finalizing the Graph
Combine all plotted points onto a single graph using different colors for each equation. Label each spiral for clarity and verify that each corresponds to its respective mathematical expression.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Spirals
Spirals are captivating curves that can elegantly coil around a center point, increasing or decreasing in size depending on their definition. They appear frequently in nature, art, and science. In the context of polar coordinates, spirals can be neatly managed and expressed through equations that relate the radial distance to the angle.
- The Archimedean spiral is a classic example of a spiral where the radial distance grows linearly with the angle. This is represented as: \( r = \theta \).
- The logarithmic spiral increases its distance exponentially as the angle grows, written as: \( r = e^{\theta / 10} \). This spiral tightens as it expands.
- A hyperbolic spiral behaves oppositely, with decreasing distance as the angle increases, such as: \( r = 8/\theta \).
- Each type of spiral offers a unique view of how curves operate within geometry and the phenomenal variety of structures we can model mathematically.
Graphing Equations
Graphing equations in polar coordinates provides a dynamic way to examine mathematical relationships where the distance and angle define a curve or path. To graph an equation in polar coordinates:
- Calculate points by picking various values of \(\theta\), and compute the corresponding \(r\) values.
- Plot these points on a polar grid, which is divided into circles of equal radii and lines spreading from the center at regular angular intervals.
- The resultant curve can be smooth and continuous, representing the equation's relationship between \(r\) and \(\theta\).
Coordinate Systems
Coordinate systems are crucial for locating points in space, and polar coordinates offer a unique alternative to the commonly used Cartesian system. In polar coordinates:
- Each point is defined by a distance \(r\) from a reference point (the origin), and an angle \(\theta\) from a fixed direction (the positive x-axis).
- This system is particularly beneficial for detailing spirals, circles, or any figure radiating from a center point.
Mathematical Modeling
Mathematical modeling involves using mathematical constructs like equations and functions to represent real-world phenomena. In exploring spirals through polar coordinates, we can create models that
- Investigate growth patterns in biology, such as the arrangement of leaves or the shells of mollusks.
- Simulate phenomena in physics, like the spiral paths of objects in gravitational fields.
- Design architectural elements, employing curves that naturally guide the eye.
