Determine the Area of a Rectangle with Given Perimeter and Dimensions
In this article, we delve into the process of determining the area of a rectangle given its perimeter and the relationship between its length and width. We will explore three different scenarios and their respective solutions, providing a comprehensive guide through algebraic manipulation and geometric principles.
Scenario 1: When the Length is 4 Times the Width
Consider a rectangle with a length that is 4 times its width. The perimeter is given as 45 inches. We seek to find the area of the rectangle.
Step 1: Express the length in terms of the width. Let L 4W.
Step 2: Apply the perimeter formula: P 2W 2L.
Step 3: Substitute the known perimeter and the relationship between length and width.
45 2W 4W
45 6W
Step 4: Solve for W.
W 4.5 inches
Step 5: Find the length using the relationship L 4W.
L 4 * 4.5 18 inches
Step 6: Calculate the area of the rectangle.
A LW 18 * 4.5 81 square inches
Scenario 2: When the Length is 4 Inches Greater than the Width
In this scenario, we have a rectangle where the length is 4 inches more than the width. The perimeter is given as 24 centimeters.
Step 1: Establish the relationship: L W 4.
Step 2: Apply the perimeter formula: P 2W 2L.
Step 3: Substitute the known perimeter and the relationship between length and width.
24 2W 2(W 4)
24 2W 2W 8
24 4W 8
Step 4: Simplify the equation.
24 - 8 4W
16 4W
Step 5: Solve for W.
W 4
Step 6: Find the length using the relationship L W 4.
L 4 4 8
Step 7: Calculate the area of the rectangle.
A LW 8 * 4 32 square centimeters
Scenario 3: When the Length is Twice the Width
In this case, we have a rectangle where the length is twice the width, and the perimeter is known. Let the width be w and the length be 2w.
Step 1: Express the length in terms of the width: L 2W.
Step 2: Apply the perimeter formula: P 2W 2L.
Step 3: Substitute the known perimeter and the relationship between length and width.
24 2W 2(2W)
24 2W 4W
24 6W
Step 4: Solve for W.
W 4
Step 5: Find the length using the relationship L 2W.
L 2 * 4 8
Step 6: Calculate the area of the rectangle.
A LW 8 * 4 32 square centimeters
Conclusion
By exploring these different scenarios, we have demonstrated how to determine the area of a rectangle based on its perimeter and the relationship between its length and width. Understanding these methods is crucial for solving a wide range of geometric problems in both academic settings and practical applications.