Problem 16 - Basic Technical Mathematics with Calculus

Question

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

Step-by-Step Solution

Answer

The area of the rectangle is 17892 cm².

Step 1: Understand the Formula

The area of a rectangle can be calculated using the formula: \( \text{Area} = \text{length} \times \text{width} \). This formula is derived from multiplying the rectangle's two perpendicular sides.

Step 2: Substitute the Given Values into the Formula

We have the length \( l = 142 \text{ cm} \) and the width \( w = 126 \text{ cm} \). Substituting these values into the formula, we have: \[ \text{Area} = 142 \text{ cm} \times 126 \text{ cm} \].

Step 3: Calculate the Area

Now, multiply the numbers: \( 142 \times 126 = 17892 \). Hence, the area of the rectangle is \( 17892 \text{ cm}^2 \).

Key Concepts

Geometry

Geometry is a branch of mathematics that deals with the shapes, sizes, and properties of figures. In this particular exercise, we are focused on rectangles, which are a fundamental shape in geometry. A rectangle is a quadrilateral with four right angles, meaning each corner is a 90-degree angle. The opposite sides of a rectangle are equal in length and parallel.
Understanding the properties of rectangles is crucial for calculating different measurements, such as perimeter and area. These measurements help in real-world applications where knowing the size of an object is important, like tiling a floor or painting a wall.

Measurement Formula

When it comes to determining the area of a rectangle, a specific measurement formula is used: \( \text{Area} = \text{length} \times \text{width} \). Let's break down this formula to understand how it works.

• Length: This is the measurement of the longer side of the rectangle. In our exercise, this is given as 142 cm.
• Width: This is the measurement of the shorter side of the rectangle. Here, it is given as 126 cm.

By multiplying the length by the width, we can find the area, which tells us how much surface the rectangle covers. The result is expressed in square centimeters  this is a standard way to denote area in geometry.

Mathematics Problem Solving

Solving mathematics problems requires a clear step-by-step approach, similar to the one demonstrated in solving the area of a rectangle. Let's go through the solution process together:

• Understand the problem: We need to find the area of a rectangle, which requires knowing both the length and width.
• Apply the formula: Once we know our length and width (142 cm and 126 cm), we can substitute these into the area formula \( \text{Area} = 142 \times 126 \).
• Calculate and interpret: Perform the multiplication to get the result, \( 142 \times 126 = 17892 \). This means the rectangle covers an area of \( 17892 \text{ cm}^2 \).

The key to successfully solving math problems is to break them down into smaller, manageable steps. This process makes it easier to understand and reduces the chance of errors.