Problem 15 - Basic Technical Mathematics with Calculus

Question

Find the area of each figure. Rectangle: \(l=8.35\) in. \(, w=2.81\) in.

Step-by-Step Solution

Answer

The area of the rectangle is 23.4535 square inches.

Step 1: Identify the Formula for Area

To find the area of a rectangle, we use the formula: \( A = l \times w \), where \( l \) is the length and \( w \) is the width of the rectangle.

Step 2: Substitute the Given Values

Substitute the given length and width into the formula. Here, \( l = 8.35 \) inches and \( w = 2.81 \) inches. So, \( A = 8.35 \times 2.81 \).

Step 3: Perform the Multiplication

Multiply 8.35 by 2.81 to calculate the area. This computation can be done using a calculator: \( 8.35 \times 2.81 = 23.4535 \).

Step 4: Express the Final Answer

The area of the rectangle is \( 23.4535 \) square inches. Make sure to include the unit of measurement.

Key Concepts

Geometry: Understanding Shapes and Spaces

Geometry is a branch of mathematics that deals with shapes and the spaces they occupy. It covers a broad range of concepts from basic lines and points to elaborate shapes such as polygons and polyhedra. When we talk about rectangular shapes, like in the exercise above, we’re delving into one of the simplest geometrical forms. A rectangle is defined by having four sides with opposite sides being equal and parallel. This symmetry and order make it straightforward to calculate their properties, like area and perimeter.

To fully understand geometry, it can be helpful to visualize or even draw each shape you're working with. This helps you see the properties that define them. For example, in the case of the rectangle of the exercise, you would draw a four-sided figure, ensuring that the opposite sides are equal. Hence, this helps to reaffirm the given dimensions and provides a clearer pathway for solving for its area or other properties.

Mathematics: The Language of Numbers

Mathematics is the scientific study of numbers, operations, and equations. It serves as the foundational language through which we interpret and understand the world. In calculating the area of a rectangle, we engage with mathematics, employing core principles to arrive at logical solutions. Mathematics is structured and follows rules that, if adhered to, guide you to the correct solution each time.

Using mathematical operations like multiplication in this exercise, we take the provided dimensions of length and width and effectively use the formula for area:

• The operation is direct and straightforward: Multiply the length by the width.
• As with many problems in mathematics, understanding and applying the formula correctly is crucial for solving the problem.

This exercise highlights how mathematics simplifies and solves real-world problems with precision, by providing a reliable method to find the area of a rectangle through a simple multiplication operation.

Rectangular Area Calculation: Putting it into Practice

Calculating the area of a rectangle is, in essence, about understanding and applying the correct formula:

• Start with the formula: \( A = l \times w \).
• Identify and substitute the known values, which represent the length and width of the rectangle.
• In this exercise, \( l = 8.35 \) inches and \( w = 2.81 \) inches. Insert these numbers into the formula to get \( A = 8.35 \times 2.81 \).
• Once values are substituted, perform the multiplication to find the area: \( 8.35 \times 2.81 = 23.4535 \).
• Always express the final result with the correct units. Here, it’s squared inches (\( \text{in}^2 \)) because area is a two-dimensional measurement.

The process doesn’t stop with the calculation; understanding why we find an area by multiplying and using units like square inches will deepen your comprehension of both the matter at hand and the broader applications of this knowledge.

By practicing these calculations, you enhance not only your mathematical skills but also your problem-solving abilities and readiness to tackle similar problems in more complex scenarios.