Solving for the Width of a Rectangular Field: A Comprehensive Guide

Understanding the geometry of a rectangle is crucial in various applications, from basic math problems to real-world scenarios such as land measurement. This guide provides a step-by-step solution to determine the width of a rectangular field when the perimeter and length are known.

Introduction to Rectangle Perimeter and Length/Width Calculation

The perimeter of a rectangle is the total distance around the shape. For a rectangle, the perimeter ( P ) can be calculated using the formula: [ P 2L 2W ] where ( L ) is the length and ( W ) is the width. This formula stems from adding the lengths of all four sides of the rectangle.

Solving for Width with Given Perimeter and Length

In many real-world situations, you might need to find the width of a rectangle when the perimeter and length are provided. Let's solve for the width of a rectangular field where the perimeter is 356 yards and the length is 84 yards.

Step-by-Step Solution

Identify Given Values: Perimeter (( P )) 356 yards Length (( L )) 84 yards Substitute Values into the Formula: [ 356 2 times 84 2W ] Simplify and Isolate ( W ): First, compute the term ( 2 times 84 ): [ 356 168 2W ] Subtract 168 from both sides: [ 356 - 168 2W ] Calculate the difference: [ 188 2W ] Divide both sides by 2 to solve for ( W ): [ W frac{188}{2} 94 text{ yards} ]

Thus, the width of the field is 94 yards.

Additional Examples and Proofs

Let's verify this solution with a few more examples and methods:

Example 1

Perimeter 285 yards, 2 Length 280 yards

Step-by-Step Calculation: Let breadth ( x ) Perimeter 2(Length Breadth) [ 285 2(280 x) ] [ frac{285}{2} 280 x ] [ 142.5 280 x ] [ x 142.5 - 280 ] [ x -137.5 ] Since this result is negative, we need to recheck our perimeter and length values.

Example 2

Perimeter 356 yards, Length 84 yards

Calculation: [ 356 2(84 W) ] [ frac{356}{2} 84 W ] [ 178 84 W ] [ W 178 - 84 ] [ W 94 text{ yards} ]

Conclusion

By using simple algebraic manipulation and the fundamental formula for the perimeter of a rectangle, we can accurately find the width when the perimeter and length are known. Understanding this concept can be crucial in various applications, from sports fields to real estate.

References

MathIsFun - Rectangle Perimeter Khan Academy - Rectangle Perimeter

Feel free to use this method in your studies or professional work to ensure accuracy in calculating widths and lengths of rectangles.