Problem 18 - Algebra for College Students

Question

Solve each of the following for the indicated variable. \(A=l w\) for \(l\) (Area of a rectangle)

Step-by-Step Solution

Answer

\( l = \frac{A}{w} \)

Step 1: Understand the Equation

The equation given is for the area of a rectangle, which is represented as \( A = lw \), where \( A \) is the area, \( l \) is the length, and \( w \) is the width of the rectangle. Our task is to solve for \( l \), the length.

Step 2: Isolate the Variable

To find \( l \), we need to isolate it on one side of the equation \( A = lw \). We can do this by dividing both sides of the equation by \( w \), the width. This gives us the formula \( l = \frac{A}{w} \), which expresses the length in terms of the area and the width.

Step 3: Confirm the Solution

After isolating \( l \), the formula \( l = \frac{A}{w} \) should make logical sense: the length is the area divided by the width. This formula is now correctly solved for the variable \( l \).

Key Concepts

Area of a Rectangle

The concept of the area of a rectangle is foundational in both geometry and algebra. It tells us about the space contained within a rectangle's boundaries. To calculate this area, one multiplies the rectangle's length (denoted as \( l \)) by its width (denoted as \( w \)). The resulting expression is given by the formula: \[ A = lw \] where \( A \) is the area. This simple formula allows us to determine how much surface a rectangle covers. Understanding this concept helps solve numerous real-world problems, from calculating floor spaces to determining the amount of paint needed for a wall. Knowing the area allows us to rearrange this formula to solve for other unknowns, such as length or width, if the other dimensions are known.

Isolating Variables

Isolating a variable is a crucial skill in algebra. It involves rearranging an equation to solve for a specific variable. In the equation \( A = lw \), if we are tasked to find the length \( l \), we must isolate it. This is done by manipulating the equation to express \( l \) in terms of \( A \) and \( w \). We need to get \( l \) alone on one side of the equation:

• Start with the formula for the area: \( A = lw \).
• Divide both sides by \( w \) to isolate \( l \): \( l = \frac{A}{w} \).

By dividing each side of the original equation by \( w \), we effectively cancel out \( w \) on the right side, leaving \( l \) by itself. This manipulation shows that the length is equal to the area divided by the width.

Length and Width Calculations

Length and width calculations are essential for practical applications. With the formula \( l = \frac{A}{w} \), we can determine the length of a rectangle when its area and width are known. This calculation is directly derived from the basic principle of isolating variables.Consider a problem where we know a rectangle's area is 50 square units, and its width is 5 units. Using our derived formula:

• Substitute \( A = 50 \) and \( w = 5 \) into the formula \( l = \frac{A}{w} \).
• This gives \( l = \frac{50}{5} \).
• Calculate to find \( l = 10 \) units.

This method is helpful for solving complex design or architecture problems, where specific dimensions are required but not all dimensions are directly known. Understanding how to manipulate the formula and substitute known values allows us to find unknown measurements efficiently.