Problem 67 - Elementary Algebra Within Reach

Question

In Exercises 67 and 68, write and simplify expressions for (a) the perimeter and (b) the area of the rectangular sandboxes. \(4 x \mathrm{ft}\) $$ (x+5) \mathrm{ft} $$

Step-by-Step Solution

Answer

(a) The perimeter of the rectangle is \(10x+10\) ft. (b) The area of the rectangle is \(4x^2+20x\) ft.

Step 1: Write Expressions for the Perimeter of the Rectangle

The formula for the perimeter of a rectangle is \(2(l+w)\), where \(l\) is the length of the rectangle and \(w\) is the width of the rectangle. Here, the length is \((x+5)\) ft, and the width is \(4x\) ft. Substituting these values into the formula gives \(2[(x+5)+4x]\).

Step 2: Simplify the Expressions for the Perimeter of the Rectangle

To simplify \(2[(x+5)+4x]\), distribute the \(2\) to all terms inside the brackets. Therefore, \(2[(x+5)+4x]\) simplifies to \(2x+10+8x\), which further simplifies to \(10x+10\) ft.

Step 3: Write Expressions for the Area of the Rectangle

The formula for the area of a rectangle is \(l \times w\), where \(l\) is the length of the rectangle and \(w\) is the width of the rectangle. Here, the length is \((x+5)\) ft, and the width is \(4x\) ft. Substituting these values into the formula gives \((x+5)(4x)\).

Step 4: Simplify the Expressions for the Area of the Rectangle

To simplify \((x+5)(4x)\), distribute the \(4x\) to all terms inside the brackets. Therefore, \((x+5)(4x)\) simplifies to \(4x^2+20x\) ft.

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a key skill in mathematics. It involves rewriting expressions to make them easier to work with. Here, we simplify expressions for the perimeter and area of a rectangular sandbox.
To simplify the perimeter expression:

• Start with the formula. The perimeter of a rectangle is given by the expression \(2(l+w)\).
• The length \(l\) is \((x+5)\) feet, and the width \(w\) is \(4x\) feet.
• Substitute these values into the formula: \(2[(x+5)+4x]\).
• Distribute the 2 across the terms: \(2(x+5) + 2(4x)\).
• This simplifies to: \(2x + 10 + 8x\).
• Finally, combine like terms to get \(10x + 10\) feet.

For the area of the rectangle, start with \((x+5)(4x)\):

• Expand the expression by distributing \(4x\) across \((x+5)\).
• This gives \(4x^2 + 20x\).

Hence, the simplified area of the rectangle is \(4x^2 + 20x\) square feet. Simplifying expressions helps make calculations easier and clearer, which is especially useful in solving geometry problems.

Rectangle Properties

Understanding properties of rectangles is essential in geometry. Rectangles are four-sided shapes with opposite sides that are equal in length and four right angles.
Some important properties of rectangles:

• The opposite sides are equal in length.
• Each angle in a rectangle is a right angle, meaning it's 90 degrees.
• The diagonals are equal in length and bisect each other.

When calculating perimeter and area:

• The perimeter is the total distance around the rectangle. It's calculated as twice the sum of the length and width: \(2(l+w)\).
• The area measures the space inside the rectangle, calculated as the product of its length and width: \(l \times w\).

Understanding these properties makes it easier to solve problems involving rectangles, such as calculating their perimeter and area using algebraic expressions.

Educational Geometry

Geometry is a field of mathematics that focuses on the properties of shapes and their spaces. In educational settings, geometry not only teaches students about shapes but also trains them to think critically and solve problems.
Key concepts in educational geometry include:

• Shapes and figures: Students learn to identify different shapes, such as rectangles, triangles, circles, and others.
• Measurements: Understanding how to measure different aspects like perimeter, area, and volume is crucial.
• Spatial understanding: Helps in visualizing how shapes interact in space and understanding their properties.

When students learn to calculate the perimeter and area of rectangles, they apply these geometric concepts and develop a foundational understanding that is important for more advanced mathematical topics. Using exercises like simplifying algebraic expressions related to these shapes can enhance comprehension and retention.