Question

In Problems 137-146, express each statement as an equation involving the indicated variables. Area of a Rectangle The area \(A\) of a rectangle is the product of its length \(l\) and its width \(w\).

Short answer

\(A = l \times w\)

Step-by-step solution

Understand the Variables

Identify the variables given in the problem. The area of the rectangle is symbolized by the letter A, the length by l, and the width by w.

Define the Relationship

Understand that the area (A) of a rectangle is calculated by multiplying the length (l) by the width (w).

Write the Equation

Express the relationship between the area, length, and width as an equation. The equation is: \(A = l \times w\).

Key concepts

rectangle geometry

Rectangles are four-sided polygons known as quadrilaterals. They have two pairs of opposite sides that are equal in length and all four angles are right angles (90 degrees). This unique geometry makes rectangles easy to work with when calculating their properties. Understanding these basics helps in grasping how to determine other measurements like the area. Remember, each pair of opposite sides is the same length, which will be important when you work on more complex problems involving rectangles.

area formula

The area of a rectangle measures the amount of space inside the rectangle. It's calculated using the formula: \[A = l \times w \] Here, A stands for area, l for length, and w for width. This formula stems from the basic fact that area is usually found by multiplying two dimensions. For a rectangle, it is straightforward since length and width are perpendicular to each other. Whenever you know the length and width, plug these values into the formula to find the area. This formula is crucial in many practical applications like construction, design, and even crafting.

variable identification

In mathematical problems, identifying variables correctly is a key step. Variables are symbols used to represent numbers or values in equations. In this exercise, we use three variables: A (area), l (length), and w (width). When given a problem, it's essential to first define what each variable stands for. This simplifies the process of setting up and solving equations. For instance, if a problem states that a rectangle has a length of 5 units and a width of 3 units, you identify l as 5 and w as 3. Properly identifying variables helps avoid confusion and errors as you progress through problem-solving.

equation formation

Forming equations correctly is the foundation of solving algebraic problems. Equations represent relationships between different quantities. In the case of a rectangle's area, we establish that the area (A) is the product of the length (l) and width (w). Once the variables are known, form the equation by relating them properly. As captured in the exercise, the equation is: \[A = l \times w \] This equation displays how to calculate the area given the length and width. By plugging in the specific values you have identified for l and w, you can solve for A, thus finding the area. Understanding and forming such equations enable you to tackle a wide array of related mathematical problems.