Solving for the Area of Rectangular Playing Fields: Geometry and Algebra
When dealing with rectangular playing fields, the perimeter and area are two fundamental measurements that provide crucial information about the field's dimensions. Let's explore how to calculate the area given the perimeter and a relationship between the length and width of the rectangle. The provided problems and solutions will illustrate the process step-by-step.
The First Problem: Length 6 Meters Shorter than Twice the Width
Given the perimeter of a rectangular playing field is 504 meters and its length is 6 meters shorter than twice its width, we can use geometry and algebra to find the area of the field.
Step-by-Step Solution for the First Problem
Define the variables:
l length of the rectangle in meters
w width of the rectangle in meters
The perimeter of the rectangle is given by:
P 2l 2w
Given that the perimeter is 504 meters:
2l 2w 504
Divide the entire equation by 2:
l w 252
This is Equation 1.
From the problem, we know that the length is 6 meters shorter than twice the width:
l 2w - 6
This is Equation 2.
Substitute Equation 2 into Equation 1:
2w - 6 w 252
Combine like terms:
3w - 6 252
Add 6 to both sides:
3w 258
Divide by 3:
w 86
Find the length using Equation 2:
l 2(86) - 6 172 - 6 166
Calculate the area of the rectangle using the width and length:
Area (A) l × w 166 × 86 14276 square meters
The Second Problem: Ratio Between Length and Breadth is 6:2
The problem states that the perimeter of a rectangular field is 480 meters, and the ratio of the length to the breadth is 6:2. Let's solve for the area using these conditions.
Step-by-Step Solution for the Second Problem
Define the ratio of length to breadth:
Length : Breadth 6 : 2
Let L 6x and B 2x
The perimeter of the rectangle is given by:
Perimeter (P) 2L 2B
Given that the perimeter is 480 meters:
2(6x) 2(2x) 480
Simplify the equation:
480 12x 4x 16x
Solve for x:
x 480 / 16 30
Calculate the length and breadth:
Length (L) 6x 6(30) 180 meters
Breadth (B) 2x 2(30) 60 meters
Calculate the area of the rectangle:
Area (A) L × B 180 × 60 10800 square meters
Conclusion and Verification
In both problems, we used the perimeter and the given relationships between the length and width to solve for the area. The area of the first rectangular playing field is 14276 square meters, and the area of the second field is 10800 square meters. The solutions demonstrate that by carefully applying the formulas and relationships, the area of a rectangular field can be accurately determined.
Remember to always double-check your calculations by substituting the values back into the original equations to ensure the results are correct.
Keywords: Rectangular Playing Field, Perimeter, Area, Algebraic Equations