Problem 8 - Basic Technical Mathematics with Calculus

Question

Find the perimeter of each figure. $$\text { Rectangle: } l=142 \mathrm{cm}, w=126 \mathrm{cm}$$

Step-by-Step Solution

Answer

The perimeter of the rectangle is 536 cm.

Step 1: Understand the Perimeter Formula for a Rectangle

The perimeter of a rectangle can be calculated using the formula: \( P = 2(l + w) \), where \( l \) is the length and \( w \) is the width of the rectangle.

Step 2: Substitute the Given Values

Take the length \( l = 142 \) cm and the width \( w = 126 \) cm and substitute these values into the perimeter formula: \( P = 2(142 + 126) \).

Step 3: Calculate the Sum Inside the Parentheses

Compute the sum of the length and width first: \( 142 + 126 = 268 \).

Step 4: Multiply by 2 to Find the Perimeter

Double the sum from Step 3 to find the perimeter: \( P = 2 \times 268 = 536 \) cm.

Key Concepts

Understanding the Rectangle Perimeter Formula

To find the perimeter of any rectangle, we use the simple formula: \( P = 2(l + w) \). This formula consists of two primary elements: the length \( l \) and the width \( w \) of the rectangle. By adding these two dimensions together and then multiplying by 2, we essentially calculate the total distance around the rectangle.

• Length \( (l) \): This is the longer side of a rectangle.
• Width \( (w) \): This is the shorter side of a rectangle.

The multiplication by 2 reflects the fact that a rectangle has two pairs of equal sides. The opposition of each side mirrors the other, forming the full outline.

Solving Using Step-by-Step Math Solutions

Let's break down the process into a step-by-step solution to make it easier to understand.

• Step 1: Identify the dimensions provided in the problem. For our rectangle, \( l = 142 \text{ cm} \) and \( w = 126 \text{ cm} \).
• Step 2: Substitute these values into the formula. So the equation becomes: \( P = 2(142 + 126) \).
• Step 3: Calculate the sum within the parentheses: \( 142 + 126 = 268 \).
• Step 4: Multiply the total sum by 2 to obtain the perimeter: \( P = 2 \times 268 = 536 \text{ cm} \).

These steps ensure you correctly and precisely solve the problem while enforcing the concept behind each action.

Grasping Geometry Concepts

Geometry is all about shapes, sizes, and the properties of space. Understanding the perimeter is key to grasping basic geometry concepts as it relates to how we measure the boundaries or edges of geometric shapes, like rectangles.

• Perimeter: For a rectangle, the perimeter is essentially the sum of all its edges. By understanding and calculating it, we determine how much boundary the shape encloses.
• Sides Configuration: A rectangle has two pairs of parallel sides, which means its opposite sides are equal in measurement, simplifying boundary calculations.

Real-life applications of these concepts can vary from construction to design, where precise measurements are crucial.

Deepening Mathematics Education

Learning how to calculate the perimeter of a rectangle enriches your understanding of mathematics. Math education is not just about numbers and formulas; it's also about developing logical reasoning and problem-solving skills.

• Critical Thinking: Being able to follow a step-by-step approach enhances analytical thinking and facilitates problem-solving.
• Foundation for Advanced Topics: Mastering basic concepts paves the way for more complex topics in mathematics like calculus or algebra.
• Real-World Applications: Practical uses of perimeter calculations are evident in fields such as architecture, engineering, and everyday tasks requiring measurement.

Math education enables students to approach problems methodically, ensuring they can confidently tackle real-world challenges.