Problem 45 $$ \text { Change each equatio... [FREE SOLUTION]

Chapter 8: Problem 45

$$ \text { Change each equation to rectangular coordinates and then graph. } $$ $$ r(2 \cos \theta+3 \sin \theta)=6 $$

Short Answer

Expert verified

Convert to rectangular coordinates: $ 2x + 3y = 6 $, then graph this line.

Step by step solution

Understand the Relationship

The given equation is a polar equation, and we need to convert it to rectangular coordinates. In polar coordinates, a point is represented by $ (r, \theta) $, where $ r $ is the radius and $ \theta $ is the angle. In rectangular coordinates, a point is represented by $ (x, y) $. To convert, we use the relationships: $ x = r \cos \theta $ and $ y = r \sin \theta $.

Expand Multiplication

Start with the given equation: \[ r(2 \cos \theta + 3 \sin \theta) = 6 \]Distribute $ r $ to both terms inside the parenthesis: \[ 2r \cos \theta + 3r \sin \theta = 6 \]

Substitute Rectangular Coordinates

Replace $ r \cos \theta $ with $ x $ and $ r \sin \theta $ with $ y $ in the expanded equation:\[ 2x + 3y = 6 \]We have now expressed the equation in rectangular coordinates.

Rearrange for Graphing

To make graphing easier, rearrange the equation for the slope-intercept form $ y = mx + b $:\[ 3y = -2x + 6 \]Divide every term by 3:\[ y = -\frac{2}{3}x + 2 \]This is the linear equation in slope-intercept form, where the slope $ m = -\frac{2}{3} $ and the y-intercept $ b = 2 $.

Graph the Equation

To graph the line $ y = -\frac{2}{3}x + 2 $, start at the y-intercept $ (0, 2) $. From this point, use the slope; for each step of 3 units in the positive x-direction, go 2 units down, because the slope is negative. Continue plotting points and draw the line.

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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 of representing points on a plane using a radius and an angle. This system is defined by the pair $(r, \theta)$, where:

$r$ is the radial distance from the origin (center of the graph) to the point.
$\theta$ is the angle formed with the positive x-axis.

Understanding polar coordinates is especially useful for solving problems involving rotations or circular paths. Unlike rectangular coordinates, where points are defined by their horizontal and vertical distances (x and y), polar coordinates focus on the direction and magnitude. This can simplify calculations when working with curves, circles, or any elements that involve angles.

Rectangular Coordinates

Rectangular coordinates, or Cartesian coordinates, are used to identify a point on a plane with an ordered pair (x, y). Here,

$x$ represents the horizontal position from the origin.
$y$ represents the vertical position from the origin.

This system is familiar and intuitive because it aligns with the grid-like layout often used in mathematics and everyday contexts. When converting from polar to rectangular coordinates, the following formulas are used:

$x = r \cos \theta$
$y = r \sin \theta$

These expressions help bridge the gap between the two systems and enable us to represent polar equations in a more familiar format.

Graphing Linear Equations

Graphing linear equations involves plotting straight lines on a coordinate plane. A linear equation in two variables can be expressed in the standard form $ax + by = c$, which can be easily rearranged into other useful forms for graphing, such as the slope-intercept form.To graph a linear equation, you typically:

Identify the slope (m) and y-intercept (b) if the equation is in the slope-intercept form.
Plot the y-intercept on the graph.
Use the slope to determine the direction and steepness of the line. The slope indicates how many units you move vertically for a given horizontal movement.

This method guarantees that the points lie on the line described by the equation, allowing an accurate visual representation.

Slope-Intercept Form

The slope-intercept form of a linear equation is: $y = mx + b$. In this format:

$m$ is the slope, which measures the steepness and direction of the line.
$b$ is the y-intercept, the point where the line crosses the y-axis.

This form is particularly useful because it quickly reveals both the y-intercept and slope, facilitating easy graphing. We can understand the interaction between x and y at a glance. For instance, the equation $y = -\frac{2}{3}x + 2$ tells us:

The slope, $-\frac{2}{3}$, means for every 3 units moved right along the x-axis, the line drops 2 units vertically.
The y-intercept, $2$, means the line crosses the y-axis at $y = 2$.

By examining the slope-intercept form, it becomes easy to sketch the behavior of the line and understand how the variables relate to each other.

91影视

$$ \text { Change each equation to rectangular coordinates and then graph. } $$ $$ r(2 \cos \theta+3 \sin \theta)=6 $$

Short Answer

Step by step solution

Understand the Relationship

Expand Multiplication

Substitute Rectangular Coordinates

Rearrange for Graphing

Graph the Equation

Key Concepts

Polar Coordinates

Rectangular Coordinates

Graphing Linear Equations

Slope-Intercept Form

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