Problem 5 - Basic Technical Mathematics with Calculus

Question

Find the volume or area of each solid figure for the given values. See Figs. 2.109 to 2.115 . Volume of cube: \(\quad e=7.15 \mathrm{ft}\)

Step-by-Step Solution

Answer

The volume of the cube is approximately 365.17 cubic feet.

Step 1: Understand the Formula for Volume of a Cube

The volume of a cube is calculated using the formula: \( V = e^3 \) where \( e \) is the length of each edge of the cube. This formula works because a cube has all sides of equal length.

Step 2: Substitute Given Values into the Formula

We are given that the edge length \( e \) of the cube is 7.15 feet. Substituting this into the formula, we have: \( V = (7.15)^3 \).

Step 3: Calculate the Volume

Now calculate \( (7.15)^3 \). First, multiply 7.15 by itself to get 51.1225 (i.e., \(7.15 \times 7.15\)). Then multiply this result by 7.15 again to get \(51.1225 \times 7.15 = 365.170375 \).

Step 4: Round the Result

In many practical cases, it may be appropriate to round the result to a reasonable number of decimal places. Here, we will round to two decimal places, giving us \( V \approx 365.17 \) cubic feet.

Key Concepts

Cube

A cube is a fascinating three-dimensional shape that is often encountered in everyday life. It's a solid figure bounded by six identical squares, with each side meeting at right angles. This means all edges of the cube have the same length, which makes many calculations simpler.
When we talk about a cube, think of objects like dice, ice cubes, or a Rubik’s cube. These are perfect examples of cubes since they all have equal side lengths.

• Each of the six faces of the cube is a square.
• All twelve edges have the same length.
• A cube has eight vertices or corners.

Understanding the basic structure of a cube helps in visualizing and calculating its various properties, such as volume and surface area. A solid grasp of what a cube is, sets the foundation for diving into mathematical calculations involving cubes.

Formula for Volume

The formula for calculating the volume of a cube is straightforward and derives from the fundamental properties of a cube: all its edges are of equal length. When you want to find the volume, you're trying to calculate how much space the cube occupies.
To calculate the volume of a cube, use the formula:

• \[ V = e^3 \]

In this formula, \( V \) stands for the volume, and \( e \) represents the length of the edge of the cube. "Cubing" \( e \) means multiplying it by itself twice (\( e imes e imes e \)).
This formula works specifically for cubes due to their equal edges, making it possible to simply take one edge and cube it to obtain the total volume. It’s a powerful and simple mathematical tool to measure space within a cube!

Mathematics Education

Mathematics education aims to equip students with the skills to understand a variety of mathematical concepts and apply these in problem-solving. Learning about shapes like cubes and their properties can be a fantastic way to build spatial awareness and analytical thinking.
Grasping how to use formulas such as the one for calculating the volume of a cube is part of this learning process. It encourages logical reasoning as students learn to:

• Identify the components of a mathematical formula.
• Substitute values accurately.
• Perform calculations systematically.
• Understand concepts deeply rather than just memorizing formulas.

Using real-world examples and interactive teaching methods can make these concepts exciting and relatable. This approach helps students visualize concepts and see their applications, ensuring that they gain a meaningful and practical understanding of volume, making math both enjoyable and enlightening.