Problem 63 - Algebra for College Students

Question

Set up an equation and solve each problem. The combined area of a square and a rectangle is 64 square centimeters. The width of the rectangle is 2 centimeters more than the length of a side of the square, and the length of the rectangle is 2 centimeters more than its width. Find the dimensions of the square and the rectangle.

Step-by-Step Solution

Answer

The square has a side of 4 cm; the rectangle is 6 cm by 8 cm.

Step 1: Define Variables

Let the side length of the square be \( x \) cm. The width of the rectangle, which is 2 cm more than the side of the square, can be expressed as \( x + 2 \) cm. The length of the rectangle, which is 2 cm more than its width, can be expressed as \( x + 4 \) cm.

Step 2: Write the Expression for Each Area

The area of the square is \( x^2 \). The area of the rectangle is the width times the length, \((x + 2)(x + 4)\).

Step 3: Set Up the Equation

The combined area of the square and the rectangle is given as 64 square centimeters. Therefore, the equation is \( x^2 + (x + 2)(x + 4) = 64 \).

Step 4: Expand the Rectangle's Area Expression

Expand \((x + 2)(x + 4)\):\[(x + 2)(x + 4) = x^2 + 4x + 2x + 8 = x^2 + 6x + 8\]

Step 5: Simplify the Equation

Substitute \((x + 2)(x + 4) = x^2 + 6x + 8\) into the combined area equation:\[x^2 + x^2 + 6x + 8 = 64\]Simplify to get:\[2x^2 + 6x + 8 = 64\]

Step 6: Solve the Quadratic Equation

Rearrange and solve the quadratic equation:\[2x^2 + 6x + 8 = 64\]\[2x^2 + 6x - 56 = 0\]Divide the entire equation by 2:\[x^2 + 3x - 28 = 0\]Factor the quadratic:\[(x + 7)(x - 4) = 0\]So, \(x = -7\) or \(x = 4\). Since a length cannot be negative, \(x = 4\).

Step 7: Find Dimensions

The side of the square is \(x = 4\) cm. The width of the rectangle is \(x + 2 = 6\) cm. The length of the rectangle is \(x + 4 = 8\) cm.

Key Concepts

Area of a Square

The area of a square is an important concept in geometry. When finding the area of any shape, you're essentially determining how much space the shape occupies. For a square, the formula you use is quite simple because all sides of a square are equal in length. Thus,

• Area of a square = side length × side length, or simply side²

In our example, if the side of a square is represented by the variable \( x \), then the area can be calculated as \( x^2 \). Grasping this concept helps you easily apply the formula whenever you encounter a square in math problems.

Area of a Rectangle

The area of a rectangle involves slightly different calculations because a rectangle has unequal pairs of sides. The opposite sides of a rectangle are equal, which means you have a length and a width to consider. The formula to find the area is straightforward:

• Area of a rectangle = length × width

In our exercise, the width of the rectangle is slightly related to the square, being \( x + 2 \) cm. Its length is \( x + 4 \) cm. So, the area expression becomes \((x + 2)(x + 4)\). Understanding this formula is crucial when solving problems involving rectangles in real-world applications or theoretical exercises.

Solving Equations

Solving equations, particularly quadratic equations, is a fundamental skill in algebra that enables you to find unknown values. When faced with quadratic equations, you often have a standard form like \( ax^2 + bx + c = 0 \). The exercise example was especially relevant because it involved combining areas to create a quadratic equation:

• Combine expression: \( x^2 + (x + 2)(x + 4) = 64 \)

After expanding and simplifying, you arrive at a quadratic equation: \( 2x^2 + 6x - 56 = 0 \). We simplify it by dividing through by 2, resulting in \( x^2 + 3x - 28 = 0 \). The next step involves factoring, where you're looking for two numbers that multiply to the constant term (-28) and add up to the linear coefficient (3). Here, factor to \((x + 7)(x - 4) = 0\). Finally, find that \( x = 4 \), since negative lengths are not possible. Understanding this process is vital to solving various algebraic problems effectively.