Finding the Area of the Region

Oct 13, 2025 by talynt.org 118 views

Introduction

In mathematics, the area of a region bounded by lines is a fundamental concept that has numerous applications in various fields, including geometry, calculus, and physics. In this article, we will delve into the problem of finding the area of the region bounded by the lines $2x + 5y = 2$ , $x = 0$ , and $y = 0$ . We will explore the mathematical concepts and techniques required to solve this problem and provide a step-by-step solution.

Understanding the Problem

The problem asks us to find the area of the region bounded by the lines $2x + 5y = 2$ , $x = 0$ , and $y = 0$ . To approach this problem, we need to understand the geometric representation of these lines and how they intersect to form the region of interest.

Geometric Representation

The line $2x + 5y = 2$ can be rewritten in the slope-intercept form as $y = -\frac{2}{5}x + \frac{2}{5}$ . This line has a negative slope, indicating that it slopes downward from left to right. The line $x = 0$ represents the y-axis, and the line $y = 0$ represents the x-axis.

Intersection Points

To find the intersection points of these lines, we need to solve the system of equations formed by the lines. The intersection point of the line $2x + 5y = 2$ and the line $x = 0$ can be found by substituting $x = 0$ into the equation $2x + 5y = 2$ . This gives us $5y = 2$ , which implies that $y = \frac{2}{5}$ . Therefore, the intersection point of these two lines is $(0, \frac{2}{5})$ .

Similarly, the intersection point of the line $2x + 5y = 2$ and the line $y = 0$ can be found by substituting $y = 0$ into the equation $2x + 5y = 2$ . This gives us $2x = 2$ , which implies that $x = 1$ . Therefore, the intersection point of these two lines is $(1, 0)$ .

Region of Interest

The region of interest is bounded by the lines $2x + 5y = 2$ , $x = 0$ , and $y = 0$ . This region is a triangle with vertices at $(0, 0)$ , $(0, \frac{2}{5})$ , and $(1, 0)$ .

Finding the Area of the Region

To find the area of the region, we can use the formula for the area of a triangle, which is given by $\frac{1}{2}bh$ , where $b$ is the base of the triangle and $h$ is the height of the triangle.

Base and Height

The base of the triangle is the distance between the points $(0, 0)$ and $(1, 0)$ , which is equal to $1$ . The height of the triangle is the distance between the points $(0, 0)$ and $(0, \frac{2}{5})$ , which is equal to $\frac{2}{5}$ .

Area of the Triangle

Using the formula for the area of a triangle, we can calculate the area of the region as follows:

\text{Area} = \frac{1}{2}bh = \frac{1}{2}(1)\left(\frac{2}{5}\right) = \frac{1}{5}

Therefore, the area of the region bounded by the lines $2x + 5y = 2$ , $x = 0$ , and $y = 0$ is equal to $\frac{1}{5}$ square units.

Conclusion

In this article, we explored the problem of finding the area of the region bounded by the lines $2x + 5y = 2$ , $x = 0$ , and $y = 0$ . We used geometric representation, intersection points, and the formula for the area of a triangle to find the area of the region. The final answer is $\frac{1}{5}$ square units.

References

[1] "Geometry: A Comprehensive Introduction". McGraw-Hill Education.
[2] "Calculus: Early Transcendentals". Cengage Learning.
[3] "Physics for Scientists and Engineers". Cengage Learning.

Glossary

Area: The amount of space inside a region.
Base: The distance between two points on a line.
Height: The distance between two points on a line.
Region: A set of points in a geometric space.
Triangle: A polygon with three sides and three vertices.
The Area of the Region Bounded by Lines: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the problem of finding the area of the region bounded by the lines $2x + 5y = 2$ , $x = 0$ , and $y = 0$ . We used geometric representation, intersection points, and the formula for the area of a triangle to find the area of the region. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving this problem.

Q&A

Q: What is the geometric representation of the line $2x + 5y = 2$ ?

A: The line $2x + 5y = 2$ can be rewritten in the slope-intercept form as $y = -\frac{2}{5}x + \frac{2}{5}$ . This line has a negative slope, indicating that it slopes downward from left to right.

Q: How do you find the intersection points of the lines $2x + 5y = 2$ and $x = 0$ ?

A: To find the intersection point of the line $2x + 5y = 2$ and the line $x = 0$ , we need to substitute $x = 0$ into the equation $2x + 5y = 2$ . This gives us $5y = 2$ , which implies that $y = \frac{2}{5}$ . Therefore, the intersection point of these two lines is $(0, \frac{2}{5})$ .

Q: How do you find the intersection points of the lines $2x + 5y = 2$ and $y = 0$ ?

A: To find the intersection point of the line $2x + 5y = 2$ and the line $y = 0$ , we need to substitute $y = 0$ into the equation $2x + 5y = 2$ . This gives us $2x = 2$ , which implies that $x = 1$ . Therefore, the intersection point of these two lines is $(1, 0)$ .

Q: What is the formula for the area of a triangle?

A: The formula for the area of a triangle is given by $\frac{1}{2}bh$ , where $b$ is the base of the triangle and $h$ is the height of the triangle.

Q: How do you calculate the area of the region bounded by the lines $2x + 5y = 2$ , $x = 0$ , and $y = 0$ ?

A: To calculate the area of the region, we need to find the base and height of the triangle. The base of the triangle is the distance between the points $(0, 0)$ and $(1, 0)$ , which is equal to $1$ . The height of the triangle is the distance between the points $(0, 0)$ and $(0, \frac{2}{5})$ , which is equal to $\frac{2}{5}$ . Using the formula for the area of a triangle, we can calculate the area of the region as follows:

\text{Area} = \frac{1}{2}bh = \frac{1}{2}(1)\left(\frac{2}{5}\right) = \frac{1}{5}

Q: What is the final answer to the problem?

A: The final answer to the problem is $\frac{1}{5}$ square units.

Conclusion

In this article, we provided a Q&A guide to help you better understand the concepts and techniques involved in solving the problem of finding the area of the region bounded by the lines $2x + 5y = 2$ , $x = 0$ , and $y = 0$ . We hope that this guide has been helpful in clarifying any doubts you may have had about the problem.

References

[1] "Geometry: A Comprehensive Introduction". McGraw-Hill Education.
[2] "Calculus: Early Transcendentals". Cengage Learning.
[3] "Physics for Scientists and Engineers". Cengage Learning.

Glossary

Area: The amount of space inside a region.
Base: The distance between two points on a line.
Height: The distance between two points on a line.
Region: A set of points in a geometric space.
Triangle: A polygon with three sides and three vertices.