Problem 9 - Algebra 1: Concepts and Skills

Question

In Exercises 9 and 10 , use the formula for the area of a rectangle, \(A=\ell w\) Find a formula for \(w\) in terms of \(A\) and \(\ell\)

Step-by-Step Solution

Answer

A formula for width \(w\) in terms of area \(A\) and length \(l\) is: \(w = \frac{A}{l}\).

Step 1: Write down the original formula

The area of a rectangle is given by the formula: \(A=lw\).

Step 2: Isolate \(w\) in the formula

To isolate \(w\) in the formula, divide both sides by \(l\). This gives us: \(w = \frac{A}{l}\).

Step 3: Interpret the result

The width of a rectangle can be found by dividing the area of the rectangle by its length.

Key Concepts

Area of a Rectangle

The area of a rectangle is a fundamental concept in geometry. A rectangle is defined by two pairs of parallel sides, where opposite sides are equal in length. To calculate the area of a rectangle, we use the formula: \( A = \ell \times w \) where \( A \) is the area, \( \ell \) is the length, and \( w \) is the width.

• The area measures how much space is contained within the rectangle.
• This formula is key for determining the surface area of rectangular spaces.

Understanding this formula is crucial in solving real-life problems, such as finding the amount of material needed to cover a rectangular surface. As we progress into more complex shapes, the principle remains—calculate the space within a shape.

Formula Manipulation

Formula manipulation is the process of rearranging a formula to solve for a different variable. This is a valuable skill in algebra, where multiple variables are often involved. When manipulating formulas:

• Identify the variable you want to solve for (in this case, \( w \)).
• Use inverse operations to rearrange the formula.
• Maintain balance by performing the same operation on both sides of the equation.

In our example, the original formula for the area of a rectangle is \( A = \ell \times w \). To solve for \( w \), we perform the inverse operation of multiplication, which is division. So, divide both sides by \( \ell \): \[ w = \frac{A}{\ell} \] This process transforms the area formula into one that directly calculates the width when the area and length are known.

Isolating Variables

Isolating variables is a fundamental technique in solving equations. It involves rearranging the equation so that the desired variable stands alone on one side. This is particularly useful when given a formula, and you need to solve for a specific component.To isolate a variable:

• Start by identifying which variable needs to be isolated (in this exercise, it's \( w \)).
• Perform arithmetic operations to rearrange the formula.
• The goal is to have the variable by itself on one side of the equation.

In our rectangle area formula, \( A = \ell \times w \), we isolate \( w \) by dividing both sides by \( \ell \). This action clears the multiplication of \( \ell \) and results in the rearranged formula: \[ w = \frac{A}{\ell} \] This skill is not only necessary for academic exercises but is also useful in various practical applications, allowing flexibility and understanding in different scenarios.