Problem 7 - Algebra and Trigonometry

Question

Express the given quantity in terms of the indicated variable. The area (in \(\mathrm{ft}^{2} )\) of a rectangle that is three times as long as it is wide; \(w=\) width of the rectangle (in \(\mathrm{ft} )\)

Step-by-Step Solution

Answer

The area is expressed as \( A = 3w^{2} \).

Step 1: Understanding the Problem

We need to express the area of a rectangle in terms of its width, given that the length is three times the width (i.e., the length is represented as three times of width). Let's denote the width as \( w \).

Step 2: Express Length in Terms of Width

Since the length \( l \) is three times the width \( w \), we can write it as \( l = 3w \).

Step 3: Express Area Formula Using Dimensions

The area \( A \) of a rectangle is calculated by multiplying length by width. Thus, the area formula is \( A = l \times w \).

Step 4: Substitute Length in Terms of Width

Substitute \( l = 3w \) into the area formula: \( A = 3w \times w \).

Step 5: Simplify the Expression

Simplify the expression to \( A = 3w^{2} \). This is the area in terms of the width \( w \).

Key Concepts

Area of a Rectangle

The area of a rectangle is a basic geometric concept that determines how much space is contained within the shape's boundaries. It's calculated using the simple formula:

• Area (\(A\)) = Length (\(l\)) × Width (\(w\)).

To visualize this, imagine a rectangle as a wider field or a floor. The length is how "long" it is when you walk from one end to the other. The width is how "across" you are when you step sideways. Multiplying these two dimensions gives you the total space, or area, of the rectangle.
For example, if a rectangle has a length of 8 feet and a width of 4 feet, the area becomes \(A = 8 \times 4 = 32 \, \text{ft}^2\). This means there are 32 square feet inside the rectangle.

Expressing Variables

In algebra, expressing variables means writing one variable in terms of another. This is crucial when dealing with formulas and real-world problems where dimensions change.
For example, if you know one dimension of a rectangle and a relationship exists between the two dimensions, you can express one dimension as a function of the other.
In our rectangle problem, we are told the length is three times the width. Thus, if the width is 'w', the length can be expressed in terms of 'w' as:

• \(l = 3w\).

This relationship helps us calculate other properties, like the area, using fewer known values. By expressing variables efficiently, we can work through algebra problems with clarity and precision.

Simplifying Expressions

Simplifying expressions is an algebraic process where you rewrite equations or formulas in a more straightforward form without changing their values. By simplifying, you make expressions easier to understand and work with.
In our example, the area expression becames \(A = 3w \times w\). This needs simplification.

• This expression states that the area is 3 times the width times the width itself.

Notice that the width is repeated in multiplication, showing that it's squared. We express this as \(3w^2\).

• Simplifying helps highlight how each dimension contributes to the overall formula.

By understanding and simplifying expressions, you make mathematical problems more manageable, especially in problems involving multiple steps or larger equations.