Solving for the Dimensions of a Rectangular Carpet Using Algebraic Methods

Let's consider a real-world problem involving a rectangular carpet. Suppose the area of the carpet is 48 square meters, and its length is 2 meters more than its width. We aim to find the exact dimensions of the carpet using algebraic methods.

Setting Up the Problem

To tackle this problem, we begin by defining variables to represent the width and length of the carpet:

Width: Let the width of the carpet be ( w ) meters. Length: Since the length is 2 meters more than the width, we can express the length as ( w 2 ) meters.

Formulating the Equations

The area of the carpet is given by the product of its length and width. Given that the area is 48 square meters, we can write:

[text{Area} text{Length} times text{Width}] [therefore (w 2) times w 48]

Expanding this equation:

[(w 2) times w w^2 2w] [therefore w^2 2w 48]

Subtract 48 from both sides to set the equation to zero:

[w^2 2w - 48 0]

Solving the Quadratic Equation

We can solve this quadratic equation using the quadratic formula:

[text{Quadratic formula} w frac{-b pm sqrt{b^2 - 4ac}}{2a}]

For our equation ( w^2 2w - 48 0 ), the coefficients are:

a 1 b 2 c -48

First, we calculate the discriminant:

[Delta b^2 - 4ac 2^2 - 4 times 1 times -48 4 192 196]

Next, we substitute the values of ( a ), ( b ), and ( Delta ) into the quadratic formula:

[text{w} frac{-2 pm sqrt{196}}{2 times 1}] [text{w} frac{-2 pm 14}{2}]

This gives us two potential solutions:

[text{w} frac{12}{2} 6 quad text{- or -} quad text{w} frac{-16}{2} -8]

Since a negative width does not make sense in the context of this problem, we discard (-8) and accept the positive value. Therefore, the width ( w ) is 6 meters.

Finding the Length

Using the width ( w 6 ) meters, we can determine the length:

[text{Length} w 2 6 2 8 , text{meters}]

Conclusion

The dimensions of the carpet are:

Width: 6 meters Length: 8 meters

By applying algebraic methods and solving the quadratic equation, we have successfully determined the exact dimensions of the carpet, which are 6 meters in width and 8 meters in length.

Relevant Equations

Area of the carpet: ( (w 2) times w 48 ) Quadratic formula: ( w frac{-b pm sqrt{b^2 - 4ac}}{2a} ) Discriminant: ( Delta b^2 - 4ac )