Problem 58 - Precalculus: Mathematical for Calculus

Question

Find the area of the triangle that lies in the first quadrant (with its base on the \(x\) -axis) and that is bounded by the lines \(y=2 x-4\) and \(y=-4 x+20\).

Step-by-Step Solution

Answer

The area of the triangle is \(6\) square units.

Step 1: Identify the Intersection Points

To find the points where the triangle touches the axes, we need to find where the lines intersect each other as well as the axes. For the intersection of the two lines \(2x - 4 = -4x + 20\), solve for \(x\). Adding \(4x\) to both sides gives \(6x - 4 = 20\). Adding \(4\) to both sides gives \(6x = 24\). Finally, dividing by \(6\) gives \(x = 4\). Substitute back into either line equation to find \(y\). Using \(y = 2x - 4\), we find \(y = 2(4) - 4 = 4\). So, the intersection point is \((4,4)\).

Step 2: Find Intersection with the Axes

First, find where these lines intersect the \(x\)-axis (where \(y=0\)). For \(y = 2x - 4\), setting \(y = 0\) gives \(0 = 2x - 4\) or \(x = 2\). Thus, the intersection point is \((2,0)\). For \(y = -4x + 20\), setting \(y = 0\) gives \(0 = -4x + 20\) or \(x = 5\). Thus, the intersection point is \((5,0)\). Next, identify the \(y\)-intercept for each line by setting \(x = 0\). For \(y = 2x - 4\), the \(y\)-intercept is \(-4\). It does not need to be considered as it does not lie in the first quadrant. For \(y = -4x + 20\), the \(y\)-intercept is \(20\).

Step 3: Determine Triangle Vertices

From the calculations, the triangle's vertices are the points where the lines and axes intersect: \((2,0)\), \((5,0)\), and \((4,4)\). These points confirm the triangle is in the first quadrant. The base of the triangle lies on the \(x\)-axis, and the height extends vertically from the base to the point \((4,4)\).

Step 4: Calculate the Base and Height

The base of the triangle is the distance between the points \((2,0)\) and \((5,0)\), which is \(5 - 2 = 3\). The height of the triangle is the \(y\)-coordinate of the point \((4,4)\), which is \(4\).

Step 5: Calculate the Area of the Triangle

Use the formula for the area of a triangle: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). Substitute the values for the base and height: \( \text{Area} = \frac{1}{2} \times 3 \times 4 = 6 \).

Key Concepts

Triangle Area

Calculating the area of a triangle is an essential skill in analytic geometry. When a triangle is formed within the coordinate plane, determining its area involves understanding the shapes' boundaries and the points of intersections.
The formula to calculate the area of a triangle is quite straightforward: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).
Here, the 'base' refers to any one of the triangle's sides, which usually lies along a straight line in simple coordinate geometry problems, like the x-axis in this case.
The 'height' of the triangle is the perpendicular distance from the chosen base to the opposite vertex.
Let's break it down:

• Identify two vertices on one of the triangle's sides, usually on the axes, to establish the base.
• The height is then the vertical distance from this base line to the third vertex.
• Plug these values into the formula to find the area.

This understanding allows us to compute the area swiftly and correctly when given a geometric problem involving linear equations and intersections.

Intersection Points

Intersection points play a crucial role in determining the exact shape and size of a triangle formed by linear equations on a graph. Finding where lines intersect with each other or with the axes is the key step in mapping out the boundaries of geometric shapes within the coordinate plane.
The process involves solving the equations simultaneously to find points common to the lines, or points where lines meet the axes. Here's what to look for:

• To find where two lines intersect, set the expressions for \( y \) equal to each other and solve for \( x \).
• For the intersection with the axes, set \( y = 0 \) to find the \( x \)-intercept, and set \( x = 0 \) to find the \( y \)-intercept.
• These intercepts and intersection points define the vertices of shapes such as triangles in the context of the problem.

It’s like connecting the dots on a map. Once you have your points, you have a clear path to outline the triangle within the plane.

Line Equations

Line equations are the backbone of analytic geometry problems as they describe the path a line travels on a graph. The standard form used here is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Understanding line equations helps map out where lines intersect with one another or with coordinate axes, which is crucial when solving problems related to shapes within the plane.
Let's decode this:

• The slope \( m \) indicates the steepness and direction of the line.
• The y-intercept \( b \) reveals where the line crosses the y-axis.
• Analyzing these elements helps establish where exactly lines interact with each other and with the axes.

This comprehension provides the necessary insight to trace and manipulate line equations effectively when plotting geometric problems.