Problem 37 - Algebra and Trigonometry

Question

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? \(A=l w\) for \(w\)

Step-by-Step Solution

Answer

The width \(w\) in the context of this problem is equal to the quotient of the area \(A\) and the length \(l\), or \(w = \frac{A}{l}\).

Step 1: Isolate the Variable

The goal is to isolate variable \(w\). Start by dividing both sides of the equation \(A=lw\) by \(l\). This will leave \(w\) by itself on one side.

Step 2: Perform Division

By dividing both sides of the equation by \(l\), we obtain \(w = \frac{A}{l}\).

Key Concepts

Formulas

Formulas are the backbone of algebra and mathematics in general. They act as concise expressions of relationships between different variables. In the exercise provided, we encounter the formula for the area of a rectangle: \(A = l \times w\). This formula relates the area \(A\) to the length \(l\) and width \(w\) of a rectangle.
The beauty of formulas lies in their versatility. Once we grasp a formula, we can use it to solve various problems by plugging in known values and solving for unknowns. When working with formulas, always pay attention to the units involved and ensure they are compatible.

Variable Isolation

Isolating a variable is a crucial skill in algebra. It involves rearranging an equation so the variable of interest is on one side by itself. This allows us to solve for that particular variable.
In our exercise, we are asked to solve for \(w\) in the formula \(A = lw\). To isolate \(w\), we divide both sides of the equation by \(l\), resulting in \(w = \frac{A}{l}\).

• Identify which variable you need to isolate.
• Perform the opposite operation to both sides to get the variable alone.
• Check your work by substituting the found value back into the original equation, ensuring it holds true.

Remember, isolating a variable often involves inverse operations, such as division if multiplication is involved, or subtraction if addition is involved.

Geometry

Geometry is the study of shapes, sizes, and the properties of space. It is a branch of mathematics that deals with points, lines, surfaces, and solids. Our exercise focuses on a rectangle, one of the simplest geometric shapes. In geometry, a rectangle is defined by four sides with opposite sides being equal in length.
Understanding the properties of geometric shapes is fundamental, as it allows us to apply appropriate formulas and calculate important measurements like area and perimeter. Each geometric shape follows specific mathematical rules, making this field of study quite logical and systematic.

Area Calculation

Area calculation is vital in geometry and practical applications like engineering and architecture. The area of a shape is the amount of space enclosed within its boundaries. For rectangles, the area is calculated by the formula \(A = l \times w\).
When calculating the area, always ensure:

• Measurements for length and width are in the same units.
• The formula used is appropriate for the shape.
• The calculated area is expressed in square units (e.g., square meters, square inches).

By isolating the variable \(w\) in the formula \(A = lw\), we determined that \(w = \frac{A}{l}\). This expression allows us to find the width when the area and length are known. Area calculations are not only theoretical but also enhance practical problem-solving in real-world scenarios.