Rectangular Gardens: Solving for Dimensions Given Area and Perimeter

When designing a rectangular garden, it's crucial to understand how to calculate the area and perimeter of such a space. This article will walk through a step-by-step process to determine if a rectangular garden with a specified area and perimeter exists. We will explore the equations and the process of finding the dimensions using a quadratic equation.

Understand the Problem

A rectangular garden has a given area of 132 square meters (m2) and a perimeter of 46 meters (m). To verify if such a garden can exist, we will use the formulas for the area and perimeter of a rectangle.

Formulas for Area and Perimeter

The formula for the area A of a rectangle is A l times w where l is the length and w is the width.

The formula for the perimeter P of a rectangle is P 2l 2w Given:

A 132 m2 P 46 m

Set Up the Equations

Step 1: Convert the perimeter equation into a simpler form:

Given: 2l 2w 46

Divide the entire equation by 2:

l w 23

From the area formula:

l times w 132

Solve the System of Equations

Step 2: Express w in terms of l:

w 23 - l

Substitute this into the area equation:

l(23 - l) 132

Rearrange to form a quadratic equation:

l^2 - 23l 132 0

Solve the Quadratic Equation

Using the quadratic formula:

l frac{-b pm sqrt{b^2 - 4ac}}{2a}

Here, a 1, b -23, and c 132. Note: The discriminant is a key value to ensure the equation has real solutions.

Calculate the discriminant:

b^2 - 4ac (-23)^2 - 4(1)(132) 529 - 528 1

Now solve for l:

l frac{23 pm sqrt{1}}{2} frac{23 pm 1}{2}

This gives us two possible values for l:

l_1 frac{24}{2} 12

l_2 frac{22}{2} 11

Using w 23 - l, we get:

If l 12, then w 11 If l 11, then w 12

Conclusion

The dimensions of the garden can be either:

Length 12 m, Width 11 m Width 12 m, Length 11 m

Thus, a rectangular garden can indeed have an area of 132 m2 and a perimeter of 46 m.

This process clearly demonstrates how to solve the given problem using the formulas for area and perimeter of a rectangle and a quadratic equation.