Problem 47 - Precalculus

Question

Graph equation. \(r=4 \theta\)

Step-by-Step Solution

Answer

The graph is an Archimedean spiral starting at the origin, spiraling outward as \( \theta \) increases.

Step 1: Understand the Equation Type

The equation given is in polar form: \( r = 4\theta \). In polar coordinates, \( r \) represents the radius from the origin, and \( \theta \) represents the angle from the positive x-axis. Here, \( r \) is a function of \( \theta \), leading to a spiral-type graph.

Step 2: Identify Key Points

Key points in polar graphs are useful to visualize the curve. Calculate the radius for a few specific angles (\(0, \frac{\pi}{4}, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\)): \(r(0) = 4 \times 0 = 0\), \(r\left(\frac{\pi}{4}\right) = 4 \times \frac{\pi}{4} = \pi\), \(r\left(\frac{\pi}{2}\right) = 4 \times \frac{\pi}{2} = 2\pi\), \(r(\pi) = 4 \pi\), \(r\left(\frac{3\pi}{2}\right) = 6\pi\), and \(r(2\pi) = 8\pi\).

Step 3: Describe the Shape

The function \( r = 4\theta \) describes an Archimedean spiral, where the radius increases linearly as the angle increases. This means that the graph will start at the origin and continue to spiral outwards, with the spiral arms getting further apart as \( \theta \) increases.

Step 4: Drawing the Graph

Plot the calculated key points on polar graph paper. Begin with \( (0, 0) \), and mark the points for other calculated angles and radii. Connect these points smoothly to form a continuous spiral which should extend outward as \( \theta \) increases, reflecting the Archimedean spiral's behavior.

Key Concepts

Archimedean spiral

The Archimedean spiral, named after the ancient Greek mathematician Archimedes, is a beautiful and fascinating concept in polar coordinates. It is defined by equations of the form \( r = a + b\theta \), where \( a \) and \( b \) are constants, \( r \) is the radius, and \( \theta \) is the angle. In our specific exercise, we have \( a = 0 \) and \( b = 4 \), simplifying to the equation \( r = 4\theta \).

What makes the Archimedean spiral unique is how the distance between successive coils or turns of the spiral remains constant. This is because the radius increases linearly with the angle. Let’s simplify this:

• When \( \theta = 0 \), the radius \( r = 0 \), and the spiral begins at the origin.
• As \( \theta \) increases, so does \( r \), causing the spiral to expand outward in a regular fashion.
• The constant relationship between \( r \) and \( \theta \) ensures that each turn of the spiral is evenly spaced.

The Archimedean spiral can be seen in many natural phenomena and modern designs, from the pattern of sunflower seeds to innovative architectural structures.

graphing polar equations

Graphing polar equations involves plotting points that are determined by polar coordinates of the form \( (r, \theta) \). A point in polar coordinates is defined by its distance from the origin (\( r \)) and the angle (\( \theta \)) it makes with the positive x-axis.

To graph a polar equation like \( r = 4\theta \):

• Calculate \( r \) for various values of \( \theta \), as demonstrated in the exercise with values such as \( 0, \frac{\pi}{4}, \frac{\pi}{2}, \pi \), etc.
• Each \( \theta \) gives you a specific radius. For instance, if \( \theta = \frac{\pi}{2} \), then \( r = 4 \times \frac{\pi}{2} = 2\pi \).
• Once all points are calculated, plot them on polar graph paper, which features concentric circles at fixed radius increments from the origin.

Connecting these points smoothly on the graph will result in a curve that perfectly displays the behavior of the polar equation, in this case, an Archimedean spiral, expanding outward as the angle increases.

radius calculation

The calculation of the radius in polar coordinates is central to understanding how curves appear on a polar graph. In polar equations, the radius \( r \) often depends directly on the variable \( \theta \), as seen in the equation \( r = 4\theta \). Calculating the radius gives precise points for plotting on the polar grid.

Here's how radius calculation works:

• Choose a range of \( \theta \) values. Common choices could be multiples of \( \frac{\pi}{4} \) or \( \frac{\pi}{2} \) to evenly distribute the points across the graph.
• Substitute these \( \theta \) values into the equation. For example, \( r(\pi) = 4\pi \).
• The calculated radius \( r \) tells you how far from the origin to plot the point corresponding to each angle \( \theta \).

By performing these calculations before graphing, you ensure accuracy and help visualize how the curve unfolds as \( \theta \) changes, showcasing the linear growth characteristic of an Archimedean spiral.