Problem 8 - Basic Technical Mathematics with Calculus

Question

Find the perimeter of each figure. 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 is calculated using the formula \( P = 2l + 2w \), where \( l \) is the length and \( w \) is the width.

Step 2: Substitute the Given Values

Substitute the given values into the formula: \( l = 142 \) cm and \( w = 126 \) cm. So the formula becomes \( P = 2(142) + 2(126) \).

Step 3: Calculate Each Term Separately

Calculate the products separately: \( 2 \times 142 = 284 \) and \( 2 \times 126 = 252 \).

Step 4: Add the Products

Add the two results: \( 284 + 252 = 536 \).

Step 5: Write the Final Answer

The perimeter of the rectangle is 536 cm.

Key Concepts

Rectangle Geometry

Rectangles are fundamental shapes in geometry characterized by four sides and four right angles. They are part of the family of quadrilaterals and have some unique properties:

• Opposite sides of a rectangle are equal in length.
• All interior angles are right angles, measuring 90 degrees each.
• The diagonals of a rectangle are equal in length and bisect each other.

Understanding these properties helps in various calculations, including finding the perimeter and area. Remember, knowing just the length and width of a rectangle allows you to determine many of its other characteristics.

Perimeter Calculation

In geometry, the perimeter of a shape refers to the total length of its boundary. For a rectangle, this is calculated by summing the lengths of all its sides. Since rectangles have opposite sides that are equal, the formula for calculating the perimeter simplifies to:
\[ P = 2l + 2w \]
where:

• \( l \) is the length of the rectangle.
• \( w \) is the width.

This formula is derived straightforwardly from the properties of a rectangle's sides. By substituting the given dimensions into the formula, you can efficiently find the total perimeter, which in this case is the distance around the outer edge of the rectangle.

Mathematical Formulas

Mathematical formulas are expressions used to represent relationships between different quantities. They act as tools to solve problems efficiently and accurately. In the context of rectangles, the perimeter formula is derived from the fundamental properties of the shape.
Formulas like \( P = 2l + 2w \) enable quick assessments of spatial properties without needing to draw or measure each time. By substituting known values, such as the length and width of a rectangle, the perimeter can be quickly calculated. This makes formulas incredibly useful in various practical applications, from architecture to everyday problem-solving in mathematics.