Problem 62 - Thinking Mathematically

Question

Use a quadratic equation to solve this problem. The width of a rectangular carpet is 7 meters shorter than the length, and the diagonal is 1 meter longer than the length. What are the carpet's dimensions?

Step-by-Step Solution

Answer

The dimensions of the carpet are 8 meters by 1 meter.

Step 1: Identifying the Variables

Let's let \( x \) represent the length of the carpet. Then, the width of the carpet is \( x - 7 \) meters, and the diagonal of the carpet is \( x + 1 \) meters.

Step 2: Setting up the Equation

According to the Pythagorean theorem, the square of the length of the diagonal is equals the sum of the squares of the lengths of the sides of the rectangle. So, we have \((x + 1)^2 = x^2 + (x - 7)^2\).

Step 3: Simplifying the Equation

If we simplify the equation, we obtain \(x^2 + 2x + 1 = x^2 + x^2 - 14x + 49\). Subtraction of \(x^2\) from both sides yields \(2x + 1 = x^2 - 14x + 49\). Rearranging the terms, we get a quadratic equation \(x^2 - 16x + 48 = 0\).

Step 4: Solving the Quadratic Equation

We can solve this by factoring, giving us \((x - 8)(x - 6) = 0\). This leads to 2 solutions: \(x = 8\) and \(x = 6\). But \(x\) represents length and should be greater than \(x - 7\) which is width. So, we discard \(x = 6\) because it will lead to a negative width, and we are left with \(x = 8\). So the length of the carpet is 8 meters and the width is 8 - 7 = 1 meter.

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry. It relates the lengths of the sides of a right triangle. Specifically, the theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. For any right triangle with sides of length \(a\) and \(b\), and hypotenuse \(c\), the formula is:

• \(a^2 + b^2 = c^2\)

In the context of this exercise, the carpet's length and width are the two shorter sides of the triangle, while the diagonal acts as the hypotenuse. This relationship helps us set up the equation needed to find the dimensions of the carpet.

Factoring Quadratic Equations

Factoring is one of the effective methods to solve quadratic equations. A quadratic equation is typically expressed in the form \(ax^2 + bx + c = 0\). In the given problem, we simplify the equation to bring it into a familiar quadratic form: \(x^2 - 16x + 48 = 0\).

• The next step is to factor this equation. We look for two numbers that multiply to \(c\) (48), and sum to \(b\) (-16).
• These numbers are -8 and -6, giving us the factors \((x - 8)(x - 6) = 0\).

Factoring the quadratic equation allows us to find potential solutions for \(x\). This involves setting each factor equal to zero and solving for \(x\):

• \(x - 8 = 0\) which gives \(x = 8\)
• \(x - 6 = 0\) which gives \(x = 6\)

However, as noted, the solution \(x = 6\) would result in a negative width, which is not feasible.

Problem Solving in Mathematics

Problem solving in mathematics involves creating a strategy to understand and tackle the problem. Identifying what is known and what is being asked is the beginning. In the carpet problem, we:

• Define variables to represent unknown quantities. Here, \(x\) represents the length.
• Set up equations based on given relationships. We used the Pythagorean theorem to relate the carpet's sides and diagonal.
• Simplify and solve these equations. This involves operations like expanding and factoring quadratic expressions.

Effective problem solving requires careful consideration of all possible solutions, rejecting the ones that do not fit the physical context—like the negative width scenario in this case.

Geometry in Algebra

Geometry and algebra often intersect, enabling us to solve problems involving shapes using algebraic equations. The carpet exercise is a classic example:

• The physical properties of a rectangular shape are expressed through algebraic relationships.
• The lengths of the sides and the diagonal are incorporated into an algebraic equation using the Pythagorean theorem.
• Solving this equation involves algebraic techniques, such as factoring quadratic expressions.

This blend of geometry and algebra provides a versatile approach to solving real-world problems. Understanding the underlying geometric principles helps to better apply algebraic methods to effectively find solutions.