Problem 24 - Algebra and Trigonometry

Question

Write an algebraic formula for the given quantity. You may need to consult the formulas for area and volume listed on the inside front cover of this book. The area \(A\) of a rectangle whose length is 4 \(\mathrm{ft}\) more than its width \(x\)

Step-by-Step Solution

Answer

The area of the rectangle is given by the formula \( A = x^2 + 4x \).

Step 1: Identify the Elements of the Rectangle

To write the algebraic formula for the area of a rectangle, first identify the given dimensions. The length of the rectangle, according to the problem, is given as 4 feet more than its width. Let's denote the width of the rectangle by \( x \).

Step 2: Express the Length in Terms of Width

Based on the problem, the length of the rectangle can be expressed as a function of its width: \( \text{Length} = x + 4 \).

Step 3: Recall the Formula for Area of a Rectangle

The area \( A \) of a rectangle is calculated using the formula \( A = \text{Length} \times \text{Width} \).

Step 4: Substitute the Known Values into the Area Formula

Substitute the expression for length from Step 2 into the area formula. This gives \( A = (x + 4) \times x \).

Step 5: Expand the Expression

Distribute the \( x \) in the expression: \( A = x \cdot (x + 4) = x^2 + 4x \).

Step 6: Write the Final Algebraic Expression

The algebraic formula for the area of the rectangle is \( A = x^2 + 4x \).

Key Concepts

Area of a Rectangle

The concept of area is fundamental in mathematics. Especially when it comes to understanding the size of surfaces. The area of a rectangle is a significant measure that determines how much surface space the rectangle covers. To calculate the area, you multiply the rectangle’s length by its width. If a rectangle has a length of 8 meters and a width of 5 meters, its area would be 40 square meters, calculated as:\[ A = ext{Length} \times ext{Width} = 8 \times 5 = 40 \]In cases where the dimensions are not constant values, but instead variables or expressions, the calculation method remains the same. Simply replace the fixed dimensions with algebraic expressions.

• Always remember: the units of area will be in square measures, like square feet or square meters.
• The length and width must adhere to the same unit system to yield an accurate area representation.

Expansion of Expressions

In algebra, expressions can often be simplified or restructured. This process is called expansion. It's a valuable tool when working with formulas and solving equations. Expansion involves distributing terms and simplifying the expression as much as possible.For example, if you have an expression like \( (x + 4) \times x \), you can expand it by distributing the \( x \) across each term inside the parentheses:\[A = x \cdot (x + 4) = x^2 + 4x\]

• Notice how each term within the parentheses is multiplied by \( x \), helping simplify the expression into a simple quadratic form.
• This operation is crucial when you deal with complex algebraic formulas that need simplification before solving.

The process of expansion not only delineates the nature of an expression but also sets the stage for further algebraic manipulation like factoring or solving equations.

Geometry Basics

Geometry forms a core part of mathematics and helps us understand space and shape relations. At its core, geometry involves understanding shapes, their properties, and measurements such as area and perimeter.A rectangle is a basic geometric shape consisting of four sides with opposite sides equal and angles of 90 degrees.

• Understanding rectangles involve recognizing that they have two dimensions: length and width.
• Various formulas apply to rectangles, including those for area \( A = ext{Length} \times ext{Width} \), and perimeter \( P = 2 \times ( ext{Length} + ext{Width}) \).

Knowing these concepts allows you to solve practical problems, like calculating how much paint you need to cover a wall, or the amount of fencing required for a garden. Mastery of basic geometric shapes also lays the foundation for advanced geometry, where shapes become more complex, and calculus concepts, like volume and surface area, are also explored.