The Perimeter Of The Shaded Region In The Given Figure Is (A) L (B) L+ A (C) L+ 2r (D) L+2r+a​

by ADMIN 95 views

Introduction

The perimeter of a shape is the total length of its boundary. In the given figure, we have a shaded region that is enclosed by a rectangle and two semicircles. The perimeter of this shaded region is a combination of the lengths of the rectangle and the semicircles. In this article, we will explore the different options for the perimeter of the shaded region and determine the correct answer.

Understanding the Figure

The given figure consists of a rectangle with length 'l' and width 'a', and two semicircles with radius 'r'. The shaded region is the area enclosed by the rectangle and the two semicircles. To find the perimeter of the shaded region, we need to consider the lengths of the rectangle and the semicircles.

Option (A) - l

Option (A) suggests that the perimeter of the shaded region is equal to the length of the rectangle, which is 'l'. However, this option does not take into account the lengths of the semicircles. Since the semicircles are part of the shaded region, their lengths must be included in the perimeter.

Option (B) - l + a

Option (B) suggests that the perimeter of the shaded region is equal to the length of the rectangle plus the width of the rectangle, which is 'l + a'. However, this option still does not take into account the lengths of the semicircles. The width of the rectangle is not part of the perimeter of the shaded region.

Option (C) - l + 2r

Option (C) suggests that the perimeter of the shaded region is equal to the length of the rectangle plus twice the radius of the semicircles, which is 'l + 2r'. This option includes the lengths of the semicircles, but it does not take into account the width of the rectangle.

Option (D) - l + 2r + a

Option (D) suggests that the perimeter of the shaded region is equal to the length of the rectangle plus twice the radius of the semicircles plus the width of the rectangle, which is 'l + 2r + a'. This option includes the lengths of the semicircles and the width of the rectangle.

Conclusion

Based on the analysis of the figure and the options, we can conclude that the perimeter of the shaded region is equal to the length of the rectangle plus twice the radius of the semicircles plus the width of the rectangle, which is 'l + 2r + a'. This option includes the lengths of the semicircles and the width of the rectangle, making it the correct answer.

Final Answer

The final answer is option (D) - l + 2r + a.

Explanation

The perimeter of the shaded region is the total length of its boundary. To find the perimeter, we need to consider the lengths of the rectangle and the semicircles. The length of the rectangle is 'l', the width of the rectangle is 'a', and the radius of the semicircles is 'r'. The perimeter of the shaded region is equal to the length of the rectangle plus twice the radius of the semicircles plus the width of the rectangle, which is 'l + 2r + a'.

Step-by-Step Solution

  1. Identify the lengths of the rectangle and the semicircles.
  2. Consider the lengths of the semicircles and the width of the rectangle.
  3. Add the lengths of the rectangle and the semicircles to find the perimeter of the shaded region.
  4. The perimeter of the shaded region is equal to the length of the rectangle plus twice the radius of the semicircles plus the width of the rectangle, which is 'l + 2r + a'.

Key Concepts

  • Perimeter: The total length of the boundary of a shape.
  • Rectangle: A quadrilateral with four right angles and opposite sides of equal length.
  • Semicircles: Half of a circle.
  • Radius: The distance from the center of a circle to its edge.

Real-World Applications

  • Finding the perimeter of a shape is an important concept in architecture, engineering, and design.
  • Understanding the perimeter of a shape can help us determine the total length of a boundary, which is essential in various real-world applications.

Practice Problems

  • Find the perimeter of a rectangle with length 10 and width 5.
  • Find the perimeter of a circle with radius 4.
  • Find the perimeter of a semicircle with radius 3.

Conclusion

In conclusion, the perimeter of the shaded region in the given figure is equal to the length of the rectangle plus twice the radius of the semicircles plus the width of the rectangle, which is 'l + 2r + a'. This option includes the lengths of the semicircles and the width of the rectangle, making it the correct answer.

Introduction

In our previous article, we explored the concept of the perimeter of a shape and applied it to a specific figure with a shaded region. We determined that the perimeter of the shaded region is equal to the length of the rectangle plus twice the radius of the semicircles plus the width of the rectangle, which is 'l + 2r + a'. In this article, we will answer some frequently asked questions related to the perimeter of the shaded region.

Q: What is the perimeter of the shaded region in the given figure?

A: The perimeter of the shaded region is equal to the length of the rectangle plus twice the radius of the semicircles plus the width of the rectangle, which is 'l + 2r + a'.

Q: Why is the width of the rectangle included in the perimeter of the shaded region?

A: The width of the rectangle is included in the perimeter of the shaded region because it is part of the boundary of the shaded region. The shaded region is enclosed by the rectangle and the two semicircles, and the width of the rectangle is one of the sides of the rectangle that forms the boundary of the shaded region.

Q: Why is twice the radius of the semicircles included in the perimeter of the shaded region?

A: Twice the radius of the semicircles is included in the perimeter of the shaded region because the semicircles are part of the boundary of the shaded region. The radius of the semicircles is the distance from the center of the semicircles to their edges, and twice the radius is the total length of the semicircles.

Q: What is the significance of the length of the rectangle in the perimeter of the shaded region?

A: The length of the rectangle is significant in the perimeter of the shaded region because it is one of the sides of the rectangle that forms the boundary of the shaded region. The length of the rectangle is included in the perimeter of the shaded region because it is part of the boundary of the shaded region.

Q: Can the perimeter of the shaded region be calculated using other formulas?

A: No, the perimeter of the shaded region cannot be calculated using other formulas. The perimeter of the shaded region is equal to the length of the rectangle plus twice the radius of the semicircles plus the width of the rectangle, which is 'l + 2r + a'. This formula is the only correct formula for calculating the perimeter of the shaded region.

Q: What are some real-world applications of the perimeter of the shaded region?

A: The perimeter of the shaded region has several real-world applications, including architecture, engineering, and design. Understanding the perimeter of a shape is essential in determining the total length of a boundary, which is critical in various real-world applications.

Q: How can the perimeter of the shaded region be used in real-world applications?

A: The perimeter of the shaded region can be used in real-world applications such as designing buildings, bridges, and other structures. Understanding the perimeter of a shape is essential in determining the total length of a boundary, which is critical in various real-world applications.

Q: What are some common mistakes to avoid when calculating the perimeter of the shaded region?

A: Some common mistakes to avoid when calculating the perimeter of the shaded region include:

  • Not including the width of the rectangle in the perimeter
  • Not including twice the radius of the semicircles in the perimeter
  • Using incorrect formulas to calculate the perimeter
  • Not considering the length of the rectangle in the perimeter

Conclusion

In conclusion, the perimeter of the shaded region in the given figure is equal to the length of the rectangle plus twice the radius of the semicircles plus the width of the rectangle, which is 'l + 2r + a'. This formula is the only correct formula for calculating the perimeter of the shaded region. Understanding the perimeter of a shape is essential in determining the total length of a boundary, which is critical in various real-world applications.