Problem 30 - Algebra 2 and Trigonometry

Question

The length of a rectangle is 4 more than twice the width, \(x .\) Express the area of the rectangle in terms of \(x .\)

Step-by-Step Solution

Answer

The area of the rectangle in terms of \(x\) is \(2x^2 + 4x\).

Step 1: Understand the Problem

We are given that the length of a rectangle is 4 more than twice the width, which is represented by \(x\). The formula for the area of a rectangle is \( \text{Area} = \text{Length} \times \text{Width} \). We need to express this area in terms of \(x\).

Step 2: Express the Length in Terms of x

According to the problem, the length \(L\) of the rectangle is 4 more than twice the width. The width is represented by \(x\), so we express the length as:\[ L = 2x + 4 \]

Step 3: Express the Area in Terms of x

Now that we have the width as \(x\) and the length as \(2x + 4\), we can substitute these expressions into the formula for the area of a rectangle:\[ \text{Area} = (2x + 4) \times x \]

Step 4: Simplify the Area Expression

Distribute \(x\) in the area expression to simplify:\[ \text{Area} = 2x \cdot x + 4 \cdot x = 2x^2 + 4x \]

Key Concepts

Rectangles

To understand how to calculate and express the area of a rectangle, it's important to first comprehend what a rectangle is. A rectangle is a four-sided shape, which is a special type of quadrilateral. It has a couple of important characteristics:

• All angles are right angles, meaning they each measure 90 degrees.
• Opposite sides are parallel and equal in length.

With these properties, a rectangle maintains its shape even when expressed algebraically. This is crucial when diving into calculations, such as determining the area.

Area Calculation

When it comes to calculating the area of a rectangle, the formula is quite straightforward:

• Formula: \[\text{Area} = \text{Length} \times \text{Width}\]

This formula applies because the area represents the space within the rectangle's boundary.
In our exercise, the length is expressed in a variable term as "4 more than twice the width". The width is denoted as \(x\), leading us to express the length algebraically as \(2x + 4\).
Once we have both length and width in terms of the same variable, we can easily plug these values into the formula.
Ultimately, both the understanding of geometric shapes and good algebraic manipulation are necessary for accurately calculating the area of a rectangle in terms of variables.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that helps make complex mathematical problems easier to understand. This means reducing an expression to its most basic form. In terms of our rectangle's area calculation, simplifying means taking our expression for the area:

• \[\text{Area} = (2x + 4) \times x\]

and working to reduce it.
We use the distributive property to multiply through by \(x\):

• \[2x \cdot x + 4 \cdot x = 2x^2 + 4x\]

This simplification step is crucial. It converts a potentially complicated-looking expression into a straightforward quadratic expression \(2x^2 + 4x\), which is often easier to work with in further algebraic operations or evaluations.
This process is foundational in algebra and serves as a stepping stone to more advanced mathematical concepts, proving useful in any mathematical journey.