Introduction

A rectangular carpet has a specific area and perimeter. By utilizing the formulas for area and perimeter, we can find the dimensions of the carpet and subsequently, the length of its diagonal. This article will guide you through the process step by step, ensuring you understand how to solve similar problems.

Formulas for Area and Perimeter

For a rectangle, the area (A) is given by:

A L × W

Where L is the length and W is the width of the rectangle.

The perimeter (P) of a rectangle is calculated using:

P 2L 2W

To find the length of the diagonal (d), we use the Pythagorean theorem:

d √(L2 W2)

Problem Statement

Given a rectangular carpet with an area of 120 square meters and a perimeter of 46 meters, we need to determine the length of its diagonal.

Solution

Step 1: Set Up the Equations

From the area:

L × W 120

From the perimeter:

2L 2W 46

Which simplifies to:

L W 23

Step 2: Express W in Terms of L

Using the simplified perimeter equation:

W 23 - L

Step 3: Solve for L and W

Substitute W 23 - L into the area equation:

L(23 - L) 120

This results in a quadratic equation:

L2 - 23L 120 0

Step 4: Solve the Quadratic Equation

The solutions to the quadratic equation are given by:

L frac{-b ± √(b2 - 4ac)}{2a}

Where a 1, b -23, and c 120.

The discriminant is:

(-23)2 - 4(1)(120) 529 - 480 49

Thus, the solutions for L are:

L frac{23 ± √49}{2} frac{23 ± 7}{2}

Yielding:

L 15 or L 8

If L 15, then W 8; if L 8, then W 15.

Step 5: Calculate the Diagonal

Using the diagonal formula:

d √(L2 W2)

Substituting L 15 and W 8:

d √(152 82) √(225 64) √289 17 meters

Conclusion

The length of the diagonal of the rectangular carpet is 17 meters, indicating the distance between opposite corners of the carpet.

Additional Information

For those interested in similar topics, the following equations are helpful:

Mastering these formulas is essential for solving a variety of problems involving geometric shapes and their properties.