Problem 13 - 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 regular pyramid: square base of side 0.76 in. \(h=1.30\) in.

Step-by-Step Solution

Answer

The volume of the pyramid is approximately 0.251 cubic inches.

Step 1: Identify the Shape

The problem describes a regular pyramid with a square base. Regular in this context means that the base is a regular polygon, in this case, a square.

Step 2: Write the Volume Formula for a Pyramid

The volume of a pyramid is calculated using the formula: \( V = \frac{1}{3} B h \), where \( B \) is the area of the base and \( h \) is the height of the pyramid.

Step 3: Calculate the Area of the Square Base

Since the base is a square, calculate its area by squaring the length of one of its sides. Here, the side length is 0.76 inches.\[B = 0.76 \times 0.76 = 0.5776 \text{ square inches}\]

Step 4: Substitute the Values into the Volume Formula

Now substitute \( B = 0.5776 \) and \( h = 1.30 \) into the pyramid volume formula:\[V = \frac{1}{3} \times 0.5776 \times 1.30 \]

Step 5: Calculate the Volume

Perform the calculation:\[V = \frac{1}{3} \times 0.5776 \times 1.30 = 0.2509333 \text{ cubic inches}\]Rounded to three decimal places, the volume is 0.251 cubic inches.

Key Concepts

Pyramid Volume

Understanding how to find the volume of a pyramid is a crucial part of geometry. The volume represents the space inside the three-dimensional object.
For pyramids, the formula to calculate volume is:

• \( V = \frac{1}{3} B h \)

The variable \( B \) represents the area of the pyramid's base, while \( h \) is the height of the pyramid. This formula is always true for any pyramid, regardless of the shape of its base.
Calculating the volume involves two main steps: computing the area of the base and using that in the volume formula. This approach simplifies determining volumes of various pyramids, regardless of whether their base is square, rectangular, triangular, or any other polygon.

Square Base Area

Squares play a critical role in pyramid volume calculations when the base is a square. The area of a square base determines part of the overall pyramid's volume.
To find the area of a square, you multiply the length of one side by itself, often expressed as a square of the side length:

• \( B = \text{side} \times \text{side} \)

In the given exercise, the side length of the square base is 0.76 inches, which results in an area of

• \( 0.76 \times 0.76 = 0.5776 \text{ square inches} \)

Knowing this area allows you to progress with the volume formula, easily integrating these simple calculations into more complex geometry problems.

Geometry

Geometry is the branch of mathematics dealing with shapes, sizes, and the properties of space. It prominently features concepts like area, volume, and perimeter.
Geometry allows us to understand the properties of two-dimensional shapes, like squares and triangles, and three-dimensional objects such as pyramids and spheres.
By understanding its principles, we can measure tangible, real-world objects. Whether solving problems related to architecture, engineering, or everyday math, geometry plays a vital role. It provides the foundational skills needed to comprehend how shapes fit within space and interact with each other.
For example, understanding how pyramids are structured by combining simple shapes is a geometric principle. By building on basic geometric knowledge, like calculating square areas, one can solve more complex problems involving volume and spatial reasoning.