Problem 35 - Beginning and Intermediate Algebra

Question

Solve each problem. When appropriate, round answers to the nearest tenth. The diagonal of a rectangular rug measures \(26 \mathrm{ft},\) and the length is \(4 \mathrm{ft}\) more than twice the width. Find the length and width of the rug.

Step-by-Step Solution

Answer

Width is 10 ft, length is 24 ft.

Step 1: Understand the problem

We need to find the length and width of a rectangular rug given that the diagonal is 26 ft, and the length \( L \) is 4 ft more than twice the width \( W \).

Step 2: Set up the equations

1. First, use the Pythagorean theorem: \( L^2 + W^2 = 26^2 \). 2. Express the length in terms of the width: \( L = 2W + 4 \).

Step 3: Substitute the length in the first equation

Substitute \( L = 2W + 4 \) into \( L^2 + W^2 = 26^2 \): \[ (2W + 4)^2 + W^2 = 26^2 \]

Step 4: Simplify and solve for width

Expand and simplify the equation: \[ 4W^2 + 16W + 16 + W^2 = 676 \] Combine like terms: \[ 5W^2 + 16W + 16 = 676 \] Subtract 676 from both sides: \[ 5W^2 + 16W - 660 = 0 \] Use the quadratic formula \( W = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 5 \), \( b = 16 \), and \( c = -660 \) to solve for \( W \).

Step 5: Calculate the quadratic formula

Compute the discriminant: \[ \Delta = b^2 - 4ac = 16^2 - 4(5)(-660) = 256 + 13200 = 13456 \] Then solve for \( W \): \[ W = \frac{-16 \pm \sqrt{13456}}{2(5)} = \frac{-16 \pm 116}{10} \] This gives two solutions: \[ W = 10 \mathrm{ft} \] or \[ W = -13.2 \mathrm{ft} \]. Since width cannot be negative, \( W = 10 \mathrm{ft} \).

Step 6: Find the length

Now that we have the width, find the length: \[ L = 2W + 4 = 2(10) + 4 = 24 \mathrm{ft} \].

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental concept in geometry, especially useful for solving problems involving right triangles. It states that for any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The formula is \(a^2 + b^2 = c^2\).
In this problem, the diagonal of the rectangular rug acts as the hypotenuse, with the length and width of the rug acting as the other two sides. This allows us to use the Pythagorean theorem to form the equation \(L^2 + W^2 = 26^2\). By substituting the given expressions and solving, we find the dimensions of the rug.

rectangular dimensions

Understanding the dimensions of a rectangle is crucial for solving many geometric and algebraic problems. A rectangle is defined by its length and width.
In this exercise, the length of the rug is 4 feet more than twice the width, represented as \(L = 2W + 4\). This relationship helps us translate geometrical features into algebraic expressions. By combining this with the Pythagorean theorem, we can find both length and width, solving for one dimension when the other is known.

quadratic formula

The quadratic formula is essential for solving quadratic equations of the form \(ax^2 + bx + c = 0\). The formula is given by \(W = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
In this scenario, we formed a quadratic equation by substituting the expression of length into the Pythagorean theorem. After simplifying, we obtained \(5W^2 + 16W - 660 = 0\). Here, \(a = 5\), \(b = 16\), and \(c = -660\). We then used the quadratic formula to solve for \(W\), finding the width of the rug by computing the discriminant and solving the equation.

algebraic problem-solving

Algebraic problem-solving involves translating a word problem into mathematical equations and solving them systematically. This process includes:

• Identifying given information.
• Formulating equations based on relationships among variables.
• Applying algebraic techniques to simplify and solve the equations.

In the context of the rug problem, we identified how the length relates to the width and set up relevant equations using the Pythagorean theorem. We then solved the quadratic equation to find the width and subsequently the length. This step-by-step approach ensures we accurately translate and solve real-world problems.