Problem 9 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

$$ \text { For Problems } \text { find the equation of the line with the given characteristics. } $$ $$ \text { Passing through the origin and perpendicular to } \pi x-\sqrt{3} y=12 $$

Step-by-Step Solution

Answer

The equation of the line passing through the origin and perpendicular to \(\pi x - \sqrt{3} y = 12\) is \( y = -\frac{\sqrt{3}}{\pi} x\).

Step 1: Find the Slope of the Given Line

First, convert the equation \(\pi x - \sqrt{3} y =12\) to the slope-intercept form (y = mx + b) to find the slope (m). This can be done by isolating y. So, \(\sqrt{3} y = \pi x - 12 \), which simplifies to \(y = \frac{\pi}{\sqrt{3}} x - \frac{12}{\sqrt{3}}\) . Therefore, the slope of the given line is \(\frac{\pi}{\sqrt{3}}\).

Step 2: Find the Negative Reciprocal of the Slope

Since the line we are looking for is perpendicular to the given line, its slope is the negative reciprocal of the slope of the given line. The negative reciprocal of \(\frac{\pi}{\sqrt{3}}\) is \(-\frac{\sqrt{3}}{\pi}\) . Therefore, the slope of the line to be found is \(-\frac{\sqrt{3}}{\pi}\).

Step 3: Formulate the Equation of the Line

The slope of the line to be found is \(-\frac{\sqrt{3}}{\pi}\) and it is given that the line passes through the origin, so its y-intercept is 0. Therefore, the equation of the line in the slope-intercept form (y = mx + b) is \(y = -\frac{\sqrt{3}}{\pi} x + 0\), which simplifies to \(y = -\frac{\sqrt{3}}{\pi} x\) .

Key Concepts

Slope-Intercept Form

The slope-intercept form is a simple way to express the equation of a line. It's given by the formula \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept. This means that when \(x = 0\), the value of \(y\) is \(b\). This form is incredibly useful because it directly shows us how steep a line is and where it crosses the y-axis.

To convert a linear equation to this form, you typically need to isolate \(y\) on one side of the equation. For example, take the equation \(\pi x - \sqrt{3} y = 12\). By rearranging it to solve for \(y\), you will end up with \(y = \frac{\pi}{\sqrt{3}} x - \frac{12}{\sqrt{3}}\). Now, it's easy to see that the slope \(m\) is \(\frac{\pi}{\sqrt{3}}\) and the y-intercept \(b\) is \(-\frac{12}{\sqrt{3}}\).

• The slope \(m\) tells you how much \(y\) increases or decreases as \(x\) increases by 1.
• The y-intercept \(b\) tells you where the line crosses the y-axis.

Understanding this form makes it straightforward to analyze and graph linear equations.

Equation of a Line

The equation of a line defines a straight line on the coordinate plane. Once you have the slope \(m\) and the y-intercept \(b\), you can easily form this equation using the slope-intercept form \(y = mx + b\). This equation not only helps in graphing the line but also in identifying the relationship between any two points along the line.

In our exercise example, the line we need to find must pass through the origin and be perpendicular to another given line. The origin, which is the point (0,0), tells us that \(b = 0\). Using the slope from the previous calculation, the equation of our new line becomes \(y = -\frac{\sqrt{3}}{\pi} x\). This equation describes a line that rotates around the origin and is oriented in a specific way relative to the axis.

• Every point \((x, y)\) on this line satisfies the equation.
• Changing the slope \(m\) would tilt the line at a different angle.
• Changing the y-intercept \(b\) would move the line up or down without changing its tilt.

Recognizing these changes helps in customizing and understanding the line's behavior on the graph.

Slope of a Line

The slope of a line is a measure of its steepness and direction. It is calculated as the change in \(y\) over the change in \(x\), commonly denoted as \(m\). In simple terms, the slope tells how many units \(y\) changes for a single unit change in \(x\). If the line is ascending from left to right, \(m\) is positive; if it's descending, \(m\) is negative.

For lines that are perpendicular to each other, there's a special relationship between their slopes: the product of their slopes is \(-1\). This results because one line's slope is the negative reciprocal of the other's. For instance, the slope of the line \( \pi x - \sqrt{3} y = 12\) is \( \frac{\pi}{\sqrt{3}}\), and the slope of the perpendicular line is \(-\frac{\sqrt{3}}{\pi}\). Multiplying them, \(\frac{\pi}{\sqrt{3}} \times -\frac{\sqrt{3}}{\pi} = -1\).

• A slope of 0 means the line is horizontal.
• An undefined slope indicates a vertical line.
• Changes in slope directly affect the line's angle on the graph plane.

Understanding slopes is fundamental to analyzing how lines interact and are positioned on a graph.