Problem 8 - College Algebra

Question

Find the area of a parallelogram bounded by the \(x\) -axis, the line \(g(x)=2,\) the line \(f(x)=3 x,\) and the line parallel to \(f(x)\) passing through \((6,1)\) .

Step-by-Step Solution

Answer

The area of the parallelogram is \(\frac{34}{3}\).

Step 1: Understanding the Problem

A parallelogram is defined by four boundaries: two horizontal lines (the x-axis and another) and two lines with a given slope. We need to find the area enclosed by these boundaries.

Step 2: Identifying the Lines

The horizontal bound lines are the x-axis (y = 0) and y = 2. The line function is given as \( f(x) = 3x \) and the parallel line that passes through (6,1), which will have the same slope of 3.

Step 3: Equation of the Parallel Line

To find the equation of the line parallel to \( f(x) = 3x \) passing through (6, 1), use the point-slope form: \( y - y_1 = m(x - x_1) \). Thus, \( y - 1 = 3(x - 6) \). Simplifying, we obtain \( y = 3x - 17 \).

Step 4: Finding Intersection Points

The lines intersect with the horizontal line \( y = 2 \). Solving \( 3x = 2 \), we get \( x = \frac{2}{3} \). Solving \( 3x - 17 = 2 \), we get \( x = \frac{19}{3} \). These x-values are the intersections with y = 2.

Step 5: Calculating the Parallelogram Base

The base of the parallelogram is the distance between these intersection points, calculated as \( \left| \frac{19}{3} - \frac{2}{3} \right| = \frac{17}{3} \).

Step 6: Determining the Height

The height of the parallelogram is the vertical distance between y = 0 and y = 2, but bounded by the line equations, so it remains 2, since y = 2 is horizontal.

Step 7: Calculating the Area

The area of the parallelogram is calculated using the formula: Area = base \( \times \) height. Thus, the area = \( \frac{17}{3} \times 2 = \frac{34}{3} \).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses coordinate systems to represent and analyze geometric shapes. By converting geometric problems into algebraic equations, coordinate geometry provides a powerful tool for exploring properties like distance, slopes, and areas. You can accurately describe the position of points, lines, and shapes on a plane using ordered pairs of numbers (coordinates).

In this exercise, coordinate geometry is essential for understanding the arrangement of the parallelogram and its boundaries. The horizontal boundaries are indicated as lines parallel to the x-axis (y = 0 and y = 2). Meanwhile, the other lines are functions with slopes, indicating their angle of inclination on the plane.

Whenever you work with shapes in coordinate geometry, it's important to note the coordinates of key points such as intersections and vertices. By observing these coordinates, you can compute distances, areas, and more, which are critical for solving geometric problems step-by-step.

Point-Slope Form

The point-slope form of a linear equation allows us to construct the equation of a line when we know a point on the line and the slope. The formula is given by:\[ y - y_1 = m(x - x_1) \]

Here, \(m\) represents the slope, and \((x_1, y_1)\) are the coordinates of the given point on the line. This form is particularly useful when you need to find the equation of a line parallel or perpendicular to a given line.

In the context of the exercise, the line parallel to \( f(x) = 3x \) has the same slope, 3, as the original line. Since it passes through the point (6, 1), you substitute these values into the point-slope form to find the equation of the parallel line: \[ y - 1 = 3(x - 6) \].

After simplifying, the equation becomes \( y = 3x - 17 \), which describes the line parallel to \( f(x) = 3x \) and part of the parallelogram's boundary.

Slope-Intercept Form

Slope-intercept form is a common way to express the equation of a line. It is written as \( y = mx + b \), where \(m\) is the slope and \(b\) is the y-intercept.

This format is very intuitive as it immediately reveals the slope of the line and where it crosses the y-axis. The slope \(m\) indicates how steep the line is, while the y-intercept \(b\) specifies the point at which the line crosses the y-axis.

For example, the equation \( f(x) = 3x \) is already in slope-intercept form, showing a slope of 3 and a y-intercept of 0. Similarly, once we derived the equation \( y = 3x - 17 \) from the point-slope form, it also fits as a slope-intercept equation with slope 3 and a y-intercept of -17.

The slope-intercept form is incredibly helpful for graphing lines quickly and understanding their interaction, such as where and how they might intersect with other lines.

Intersection Points

Intersection points determine where two lines or a line and a boundary meet on the coordinate plane. Finding these points is crucial for defining the geometric shapes involved, like polygons or parallelograms.

In our parallelogram problem, to find where the lines intersect with \( y = 2 \) (the parallel line to the x-axis), you solve for \( x \) in the equations of the lines. For the line \( f(x) = 3x \), solving \( 3x = 2 \) gives \( x = \frac{2}{3} \). For the parallel line \( y = 3x - 17 \), setting it equal to 2 gives \( x = \frac{19}{3} \).

These x-values represent the locations where the lines cross \( y = 2 \), forming bases of the parallelogram. The horizontal line \( y = 2 \) intersects the inclined lines at these points, and understanding these intersections allows us to compute the length of the base accurately, which is essential for calculating the area of the parallelogram.