Problem 17 - Calculus Single Variable

Question

Find the center of \(\operatorname{mass}(\bar{x}, \bar{y})\) of the given region \(\mathcal{R}\), assuming that it has uniform mass density. $$ \begin{aligned} &\mathcal{R} \text { is the triangular region with vertices }(0,0),(0,2), \text { and }\\\ &(6,0) \end{aligned} $$

Step-by-Step Solution

Answer

The center of mass \((\bar{x}, \bar{y})\) is \((2, \frac{2}{3})\).

Step 1: Identify coordinates of vertices

The vertices of the triangular region \( \mathcal{R} \) are given as \((0,0)\), \((0,2)\), and \((6,0)\). These points represent the corners of the triangle on the xy-coordinate plane.

Step 2: Determine the base and height of the triangle

For a triangular region, one side can be recognized as the base. The base \(BC\) lies along the x-axis from \((0,0)\) to \((6,0)\), thus has length 6. The height is the y-coordinate of point \((0,2)\), which is 2.

Step 3: Calculate the area of the triangle

The area \( A \) of a triangle is given by the formula \[ A = \frac{1}{2} \text{base} \times \text{height} \]. Substituting the base and height, we get: \[ A = \frac{1}{2} \times 6 \times 2 = 6. \]

Step 4: Derive the centroid formula for a triangle

The centroid \((\bar{x}, \bar{y})\) of a triangle with vertices \((x_1, y_1) \), \((x_2, y_2) \), and \((x_3, y_3) \) is calculated as: \[ \bar{x} = \frac{x_1 + x_2 + x_3}{3}, \quad \bar{y} = \frac{y_1 + y_2 + y_3}{3}. \]

Step 5: Substituting the vertices into the centroid formula

Substitute the coordinates \((0,0)\), \((0,2)\), \((6,0)\) into the centroid formula. The x-coordinate of the center of mass is \( \bar{x} = \frac{0 + 0 + 6}{3} = 2 \). The y-coordinate is \( \bar{y} = \frac{0 + 2 + 0}{3} = \frac{2}{3} \).

Step 6: Conclude the calculation

The center of mass \((\bar{x}, \bar{y})\), which is the same as the centroid for a uniform density area, is \((2, \frac{2}{3})\).

Key Concepts

Triangular Region

A triangular region is essentially a three-sided shape enclosed by three straight-line segments. These sides are referred to as the edges, and their intersections are known as the vertices. In our exercise, the triangular region, \( \mathcal{R} \), is defined by three vertices at the points \((0,0)\), \((0,2)\), and \((6,0)\).
We can visualize this as a triangle on a coordinate plane with the base of the triangle lying along the x-axis. Triangles are fundamental geometric shapes in mathematics and are widely used in various applications, including engineering and architecture due to their structural stability. They are also frequently used in problems involving the calculation of areas and centers of mass, as seen in this exercise.

Centroid Formula

The centroid of a triangle can be thought of as its center of mass or balance point. It is also referred to as the "geometric center." The significance of the centroid lies in its uniform distribution of mass throughout the shape, assuming the triangle is homogeneous.

The formula used to find the centroid, \((\bar{x}, \bar{y})\), of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is:
\[ \bar{x} = \frac{x_1 + x_2 + x_3}{3}, \quad \bar{y} = \frac{y_1 + y_2 + y_3}{3}. \]
This formula ensures that the centroid is equidistant from all three sides of the triangle, making it a useful reference point in both theoretical and practical scenarios, such as determining the balance point of a physical object.

Coordinates of Vertices

In geometry, coordinates are used to identify specific locations on a plane. The coordinates of each vertex of a shape like a triangle are key in various calculations, including finding the area and centroid.
In our exercise, the vertices of the triangular region \( \mathcal{R} \) are given as the coordinates \((0,0)\), \((0,2)\), and \((6,0)\).

• \((0,0)\): This is the origin, where the x and y-axes intersect.
• \((0,2)\): This point lies directly above the origin, 2 units up the y-axis.
• \((6,0)\): This point is located 6 units along the x-axis from the origin.

Understanding and identifying the coordinates help in visualizing the triangle and in further calculations such as deriving the centroid and calculating the area.

Area of a Triangle

The area of a triangle is a measure of the space enclosed by the boundaries of the triangle. Calculating the area is crucial in various fields such as architecture, engineering, and everyday problem-solving.
To determine the area of a triangle, use the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height}. \]
For our triangular region \( \mathcal{R} \):
- The base is \(6\) units long, measured along the x-axis from \((0,0)\) to \((6,0)\).
- The height is \(2\) units, corresponding to the y-coordinate of the vertex at \((0,2)\).
Thus, the area \( A \) is calculated as:
\[ A = \frac{1}{2} \times 6 \times 2 = 6. \]
This mathematical concept is simple yet powerful in understanding the size and significance of the triangle in various applications.