Problem 60 - Algebra and Trigonometry

Question

Use the formula for the area of a rectangle and the Pythagorean Theorem to solve. The area of a rug is 108 square feet and the length of its diagonal is 15 feet. Find the length and width of the rug.

Step-by-Step Solution

Answer

Upon solving the equations, we find that the length and width of the rug are 12 feet and 9 feet respectively.

Step 1: Set Up the Area Equation

From the problem, we know that the area of the rug is 108 square feet. The area of a rectangle is given by the formula \(A = lw\), where \(l\) is the length and \(w\) is the width. Hence, we set up the equation \( lw = 108 \).

Step 2: Set Up the Pythagorean Theorem Equation

The Pythagorean theorem, in this context, is given by the formula \(d^{2} = l^{2} + w^{2}\), where \(d\) is the diagonal. We know that the diagonal of the rug is 15 feet. So, the equation is \(15^{2} = l^{2} + w^{2}\), which simplifies to \(225= l^{2} + w^{2}\).

Step 3: Solve the Equations

We have two equations, \(lw = 108\) and \(225= l^{2} + w^{2}\). We can rearrange the first equation as \(w = 108 / l\). Then substitute this in the second equation to obtain \(225= l^{2} + (108 / l)^{2}\). This is a quadratic equation, solve this to get the values of \(l\). Substitute the value of \(l\) in equation \(lw = 108\) to obtain \(w\).

Key Concepts

Understanding the Area of a Rectangle

The area of a rectangle is a measure of the surface covered by the rectangle. It's calculated as the product of the rectangle's length and width, using the formula \( A = lw \), where \( A \) is the area, \( l \) is the length, and \( w \) is the width.

• The units of area are always square units, which depend on the units used for length and width. For example, if both dimensions are in feet, then the area will be in square feet.
• In any rectangle, the length and width are perpendicular to each other.

The formula itself is straightforward, making it ideal for problems involving rectangular spaces, like rugs, flooring, or walls. In practical terms, knowing the area lets you determine how much material is needed for a given project, ensuring proper coverage without waste.
In problems like the original exercise, where the area is given, understanding this formula also helps in finding missing dimensions when combined with other information, such as a diagonal length.

Solving Quadratic Equations

A quadratic equation is any equation that can be written in the form \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. Quadratic equations are a key component in solving various mathematical problems, particularly those involving parabolic shapes or calculations like in our example problem when finding the dimensions of a rectangle.

• Quadratic equations can be solved using several methods: factoring, completing the square, or using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• The solution to a quadratic equation can be two distinct real numbers, a repeated real number, or two complex numbers.

In the context of the original problem, the quadratic equation arises when substituting the expression from the area equation into the Pythagorean theorem. This substitution leads to an equation that defines a relationship between the length and width when the diagonal is included. Solving this quadratic equation is crucial for finding the specific lengths that will satisfy both the area and the diagonal measurements.

Finding the Diagonal of a Rectangle

The diagonal of a rectangle is the straight line connecting two opposite vertices. It acts as a hypotenuse in right triangles formed by the height and the width of the rectangle. According to the Pythagorean Theorem, the formula for calculating the diagonal \(d\) is \( d^{2} = l^{2} + w^{2} \).

• In any rectangle, there are two diagonals, and both are equal in length.
• The diagonals' length can help in validating the rectangle's shape and dimensions.

Using the diagonal as a measure provides additional information to solve complex problems, such as finding unknown length or width when combined with the area, as in the original exercise.
For example, if you know the area and diagonal length, you can set up a system of equations to determine the rectangle's dimensions. By applying the Pythagorean Theorem with the diagonal length, you add an extra dimension to the problem-solving toolkit, making it easier to derive the necessary measurements.