Solving for the Dimensions of a Rectangular Field Using Perimeter

When dealing with rectangular fields, it is often necessary to determine the dimensions based on given information such as the perimeter. This article will walk through the process of finding the width and length of a rectangular field with a perimeter of 560 yards, where the length is four times the width.

Understanding the Problem

Let's assume the width of the field is represented as w yards. Given that the length of the field is four times the width, we can express the length as l 4w. The perimeter of a rectangle is given by the formula P 2l 2w. In this problem, the perimeter is known to be 560 yards. Therefore, we can set up the following equation:

2l 2w 560

Solving the Equation

Substituting the expression for length into the perimeter equation, we get:

2(4w) 2w 560

8w 2w 560

10w 560

w 560 / 10

w 56 yards

Now that we have the width, we can find the length by substituting w back into the expression for length:

l 4w 4 × 56 224 yards

Dimensions of the Field

The measurements for the field are in yards. Therefore, the area of the field would be in square yards, as cubic yards would be applicable to volume rather than two-dimensional area.

Alternative Methods

Another way to represent the field could be as four squares in a row. Each square would have a side length of the width w of the rectangle. This means that the perimeter can be expressed as 10 times the width:

10w 390

w 39 yards

Thus, the length would be 4 times the width:

l 4w 4 × 39 156 yards

These methods and the results confirm the dimensions of the field as 156 yards by 39 yards, with the perimeter being 390 yards.

Conclusion

By breaking down the problem into manageable steps and using the known formulas, we can efficiently solve for the dimensions of a rectangular field. Whether using the direct substitution method or considering the field as four squares, the final dimensions remain consistent, providing a clear solution.