Problem 40 - Algebra 2 and Trigonometry

Question

The width of a rectangle is \(x\) and the length is \(\sqrt{x-1}\) . If the width is twice the length, what are the dimensions of the rectangle?

Step-by-Step Solution

Answer

The dimensions of the rectangle are width = 2 and length = 1.

Step 1: Define the Problem

We are given that the width of the rectangle is \(x\), and the length is \(\sqrt{x-1}\). We need to find the dimensions of the rectangle given that the width is twice the length.

Step 2: Set Up the Equation

Since the width is twice the length, we set up the equation: \(x = 2 \times \sqrt{x-1}\). This equation relates the width and length based on the given condition.

Step 3: Isolate the Square Root

To solve the equation \(x = 2 \times \sqrt{x-1}\), first isolate the square root by dividing both sides by 2. This gives us \(\sqrt{x-1} = \frac{x}{2}\).

Step 4: Square Both Sides

To eliminate the square root, square both sides of the equation: \((\sqrt{x-1})^2 = \left(\frac{x}{2}\right)^2\). Simplifying this gives \(x-1 = \frac{x^2}{4}\).

Step 5: Rearrange and Simplify

Rearrange the equation: \(x - 1 = \frac{x^2}{4}\). Multiply through by 4 to clear the fraction: \(4x - 4 = x^2\). This simplifies to \(x^2 - 4x + 4 = 0\).

Step 6: Solve the Quadratic Equation

The equation \(x^2 - 4x + 4 = 0\) is a quadratic equation that can be factored as \((x-2)^2 = 0\). Solving this gives \(x = 2\).

Step 7: Calculate the Rectangle Dimensions

Substitute \(x = 2\) back into the expression for length to find the dimensions: \(\text{length} = \sqrt{x-1} = \sqrt{2-1} = 1\). Hence, the dimensions of the rectangle are width = 2 and length = 1.

Key Concepts

Quadratic Equations

Quadratic equations are quite significant in algebra, and their general form is \(ax^2 + bx + c = 0\). In our problem, we derived the quadratic equation \(x^2 - 4x + 4 = 0\) from the condition that the width was twice the length of a rectangle. The equation represents a parabola, and solving it involves finding the values of \(x\) that make the equation true. In this case, it's a perfect square trinomial, \((x-2)^2 = 0\), indicating that there is only one real solution: \(x = 2\). Solving quadratic equations can be done using multiple methods:

• Factoring: Break down the equation into simpler expressions. For example, \((x-2)^2 = 0\) easily shows \(x = 2\).
• Quadratic Formula: This formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), works for any quadratic equation.
• Completing the Square: A method to transform the equation into a perfect square form.

Understanding quadratic equations allows us to solve a wide array of algebraic problems, making them an essential tool in mathematics.

Rectangular Dimensions

Rectangular dimensions refer to the width and length of a rectangle, crucial for finding properties like area and perimeter. In this exercise, we examined a rectangle with specific conditions. The width \(x\) was given, while the length was expressed as \(\sqrt{x-1}\). This forms a direct relationship where knowing one dimension can help determine the other. Some key points about rectangles include:

• The Width is the shorter side or base of the rectangle, often denoted as \(b\).
• The Length is the longer side, which can be expressed in terms of width or other terms like \(\sqrt{x-1}\) in complex problems.
• Area is calculated using \(Area = width \times length\), a fundamental formula in geometry.
• Perimeter involves the sum of all sides and is given by \(P = 2(width + length)\).

Understanding rectangular dimensions is useful for solving practical geometry problems and in real-life applications, such as construction and design.

Square Root

The square root is a critical concept in algebra that involves finding a number which, when multiplied by itself, yields the original number. In mathematical terms, if \(\sqrt{n} = a\), then \(a^2 = n\). In the exercise, the length of the rectangle involved a square root expression, \(\sqrt{x-1}\), which adds complexity to the equation.Square roots have the following properties:

• It is the inverse operation of squaring - cancelling the effect of squaring a number.
• For positive numbers, the square root is always a positive number or zero.
• Square roots can be simplified if the number is a perfect square, like \(16\) or \(9\), which have square roots \(4\) and \(3\), respectively.

Understanding square roots helps in simplifying equations involving quadratic expressions and is fundamental to solving various algebraic problems.