Solving for Rectangle Dimensions Using Perimeter and Area

Often, it is a straightforward task to determine the dimensions of a rectangle when given its perimeter and area. This involves using basic algebra, but there are also simpler approaches to consider. In this article, we will explore the steps involved and present a detailed solution to the given problem.

Problem Statement: A rectangle has a perimeter of 10 inches and an area of 6 square inches. Find the length and width of the rectangle.

To begin with, we have two key equations based on the given information:

Perimeter:

2(l w) 10

l w 5

Area:

l * w 6

Our goal is to find the values of l (length) and w (width). Let's proceed step by step.

Step-by-Step Solution

1. **Simplify the Perimeter Equation:**

2(l w) 10

l w 5

2. **Solve the Area Equation:**

l * w 6

3. **Express One Variable in Terms of the Other:**

Since ( l w 5 ), we can express ( l ) in terms of ( w ):

l 5 - w

Substitute ( l 5 - w ) into the area equation:

(5 - w) * w 6

Expand and simplify:

5w - w^2 6

Rewrite the equation:

w^2 - 5w 6 0

4. **Solve the Quadratic Equation:**

w^2 - 5w 6 0

Factorize the quadratic equation:

(w - 2)(w - 3) 0

Set each factor to zero and solve for ( w ):

w - 2 0 or w - 3 0

w 2 or w 3

If ( w 2 ), then substitute back to find ( l ):

l 5 - 2 3

If ( w 3 ), then:

l 5 - 3 2

Thus, the dimensions of the rectangle can be represented as ( l 3 ) and ( w 2 ), or vice versa.

To verify, we can check the values by substituting them back into the original equations:

Perimeter Check:

2(3 2) 2 * 5 10

Area Check:

3 * 2 6

The solution satisfies both the perimeter and area constraints, confirming that the dimensions of the rectangle are ( l 3 ) inches and ( w 2 ) inches, or ( l 2 ) inches and ( w 3 ) inches.

Alternatively, it's interesting to note that ( 6 2 times 3 ) naturally fits both the area and perimeter requirements:

Area: 2 * 3 6

Perimeter: 2 * (2 3) 2 * 5 10

Thus, this is also a valid solution, and it aligns perfectly with the algebraic approach.

By understanding these methods, you can quickly determine the dimensions of any rectangle when given its perimeter and area.