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Magazine Chapter 8: Problem 93
Graph the equation using a graphing calculator. \(r=e^{\theta / 10}\) (Logarithmic spiral)
Short Answer
Expert verified
Graph the polar equation \(r = e^{\theta / 10}\) by entering it into the calculator and adjusting the viewing window.
Step by step solution
01
- Understand the Equation
The given equation is a logarithmic spiral: \[ r = e^{\theta / 10} \]Here, \( r \) represents the radius or the distance from the origin, and \( \theta \) represents the angle in radians.
02
- Set Up Your Graphing Calculator
Turn on your graphing calculator and set it to polar mode. Look for settings that allow you to enter equations in terms of \( r \) and \( \theta \).
03
- Enter the Equation
Input the equation \( r = e^{\theta / 10} \) into the graphing calculator. Make sure you use the correct syntax for exponentiation and division as per your calculator鈥檚 requirements.
04
- Adjust the Viewing Window
Adjust the viewing window to capture more of the spiral. A common setting is: \[0 \leq \theta \leq 10\pi\] and ensure the radial bounds are appropriate to see the full extent of the spiral.
05
- Graph the Equation
Plot the graph by pressing the appropriate button (usually labeled 'Graph' or similar). Observe the resulting logarithmic spiral on the screen.
06
- Analyze the Graph
Examine the graph closely to understand how the radius grows exponentially as the angle increases. Notice the pattern and shape characteristic of a logarithmic spiral.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
polar coordinates
Polar coordinates are a way to represent points on a plane using two parameters: the radius \(r\) and the angle \(\theta\). Unlike Cartesian coordinates, which use \(x\) and \(y\) axes to determine a point's position, polar coordinates focus on the distance from a central origin point and the direction from that point.
These coordinates are particularly useful for graphing spirals, circles, and other shapes that naturally extend outwards from a central point.
- The radius \(r\) measures the distance from the origin to the point.
- The angle \(\theta\) measures the rotation needed from the positive x-axis to reach the point's direction.
These coordinates are particularly useful for graphing spirals, circles, and other shapes that naturally extend outwards from a central point.
exponential growth
Exponential growth occurs when a quantity increases by a consistent percentage or factor over equal intervals of time. This type of growth is represented mathematically by functions like \(y = e^x\), where \(e\) is the base of natural logarithms.
In our equation \(r = e^{\theta / 10}\), the radius \(r\) grows exponentially as \(\theta\) increases. This means that for every unit increase in \(\theta\), the radius \(r\) increases by a factor of \(e\).
This exponential relationship is what gives the logarithmic spiral its unique shape, continually widening as it moves away from the origin.
In our equation \(r = e^{\theta / 10}\), the radius \(r\) grows exponentially as \(\theta\) increases. This means that for every unit increase in \(\theta\), the radius \(r\) increases by a factor of \(e\).
This exponential relationship is what gives the logarithmic spiral its unique shape, continually widening as it moves away from the origin.
graphing calculator usage
A graphing calculator can be an invaluable tool for visualizing complex equations like the logarithmic spiral. Here's a simple guide to using one:
Taking these steps will help you plot and analyze even the most intricate mathematical functions with ease.
- First, turn on your device and switch to polar mode. This allows equations to be entered and displayed using polar coordinates.
- Next, input the equation \(r = e^{\theta / 10}\) carefully. Make sure you follow your calculator's syntax rules for exponentiation and division.
- Since a logarithmic spiral can extend quite far, adjust the viewing window settings to cover a broad range of \(\theta\). Try \[0 \leq \theta \leq 10\pi\]
- Finally, hit 'Graph' and observe the spiral shape forming on the screen.
Taking these steps will help you plot and analyze even the most intricate mathematical functions with ease.
logarithmic spiral
A logarithmic spiral is a fascinating curve that appears frequently in nature and mathematics. This type of spiral is defined by the equation \(r = e^{\theta / 10}\).
Here are some of its key characteristics:
The logarithmic spiral is not only a mathematical curiosity but also a real-world example of nature's inherent beauty and complexity.
Here are some of its key characteristics:
- It spirals outward at an increasing rate, where the distance between turns grows exponentially.
- The angle \(\theta\) influences the rate of rotation, while the exponential factor determines how quickly the spiral expands.
- This curve can be seen in natural phenomena like seashells, hurricanes, and galaxies due to its efficient and organic growth pattern.
The logarithmic spiral is not only a mathematical curiosity but also a real-world example of nature's inherent beauty and complexity.
