Problem 16 - Basic Technical Mathematics with Calculus

Question

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

Step-by-Step Solution

Answer

The area of the rectangle is 17,892 square centimeters.

Step 1: Understand the Formula for the Area of a Rectangle

The area of a rectangle can be found using the formula: \[ \text{Area} = l \times w \]where \( l \) is the length of the rectangle and \( w \) is the width of the rectangle.

Step 2: Identify the Values

From the problem, we are given that the length \( l = 142 \) cm and the width \( w = 126 \) cm.

Step 3: Substitute the Values into the Formula

Plug the given values into the area formula: \[ \text{Area} = 142 \times 126 \]

Step 4: Calculate the Area

Perform the multiplication: \[ \text{Area} = 142 \times 126 = 17,892 \]Thus, the area of the rectangle is \( 17,892 \) square centimeters.

Key Concepts

Area Calculation

Calculating the area of a shape involves finding the surface space it covers. For rectangles and many other shapes, the process begins with understanding specific formulas.

• Knowing the formula is key if you want to find the area efficiently.
• For different shapes, the formula will differ.

Breaking it down with a tangible example like a rectangle, where you have clear dimensions, helps solidify how area calculation works. When we find the area using length and width in this context, we're determining how much two-dimensional space is within the bounds of the rectangle. Remember, the units of measurement for the area are always squared. Thus, for given dimensions in centimeters, we express results in square centimeters.

Rectangles

Rectangles are one of the most familiar geometric shapes. They have distinct properties that make calculating their area straightforward.

• Rectangles have four sides, two of which are opposite and equal in length. These are referred to as length and width.
• Each angle in a rectangle is a right angle (90 degrees).

When finding the area of a rectangle, knowing the values of the length and width is essential. This shape's simplicity, with only two dimensions to worry about, aids in easier mathematical calculations. For many real-world applications, understanding the attributes of rectangles enhances one's ability to solve practical spatial problems.

Mathematical Formulas

Mathematical formulas are the backbone of solving problems like finding the area of rectangles. The formula for the area of a rectangle is \[\text{Area} = l \times w\]where:

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

These formulas provide a standardized way to compute values, ensuring consistency across different problems. It’s crucial to understand what each variable represents to apply the formula correctly.Formulas link abstract numerical ideas with concrete real-world measurements, acting as a bridge between geometry and algebra.

Problem Solving

Problem solving in geometry often involves understanding the situation first and then applying the correct formula to find a solution. In handling problems like finding an area:

• First, identify the type of problem.
• Use the appropriate formula. In this case, \[\text{Area} = l \times w\]works for rectangles.
• Accurately substitute the known values and perform calculations.

This process cultivates a systematic approach to math problems, boosting confidence and competence in tackling more complex challenges.Following steps thoroughly ensures that there are no mistakes in your final answer, which in our example, turns out to be 17,892 square centimeters. Such skills are invaluable, making problem solving not just about finding answers but understanding the entire process.