Problem 51 - Calculus: Early Transcendentals

Question

Find the volume of the described solid \( S \). A pyramid with height \( h \) and rectangular base with dimensions \( b \) and \( 2b \).

Step-by-Step Solution

Answer

The volume of the pyramid is \( \frac{2}{3}b^2h \).

Step 1: Determine the Volume Formula for a Pyramid

The volume \( V \) of a pyramid is given by the formula:\[V = \frac{1}{3} \times \, \text{Base Area} \times \text{Height}\]

Step 2: Calculate the Base Area

The base of the pyramid is a rectangle with dimensions \( b \) and \( 2b \). The area \( A \) of a rectangle is given by:\[A = b \times 2b = 2b^2\]

Step 3: Substitute Values into the Volume Formula

Now substitute the base area \( 2b^2 \) and the height \( h \) into the pyramid volume formula:\[V = \frac{1}{3} \times 2b^2 \times h\]

Step 4: Simplify the Volume Expression

Simplify the expression to solve for the volume of the pyramid:\[V = \frac{2}{3}b^2h\]

Key Concepts

Geometry

Geometry is a branch of mathematics that explores the properties and relationships of shapes and spaces. It is all about understanding various dimensions, areas, and volumes. In the context of a pyramid, geometry helps understand how these shapes are constructed and how their dimensions relate to their volumes.

Pyramids are three-dimensional shapes consisting of a polygonal base and triangular faces converging at a single point called the apex. Whether it's a square or rectangular base, the concepts of geometry allow us to analyze and calculate the various properties of these fascinating structures. Understanding the basic geometric properties leads us to efficiently determine important measurements like the area of the base and the height, which are crucial for volume calculations.

Pyramid Formula

The formula for calculating the volume of a pyramid is fundamental in geometry. This formula defines the relationship between a pyramid's base area, height, and its resulting volume. The volume formula for a pyramid is:

• \( V = \frac{1}{3} \times \, \text{Base Area} \times \text{Height} \)

This equation highlights a couple of important facts:

• The pyramid's volume is a third of the product of its base area and height.
• The formula is derived from the idea that a pyramid is one-third the volume of a prism that has the same base area and height.

Understanding this formula allows us to calculate the space a pyramid occupies, which is crucial for applications in architecture, engineering, and various real-life scenarios.

Rectangular Base

A rectangular base is a critical element in the geometry of a pyramid. For the given problem, the base of the pyramid is a rectangle with one side being \(b\) and the other being \(2b\). The significance of knowing the shape and dimensions of the base cannot be understated:

- The base's area is needed for the volume calculation of the pyramid.- Rectangles are simple geometric shapes whose area can be effortlessly calculated.
The formula for finding the area of a rectangle is straightforward:

• \( A = \text{length} \times \text{width} \)
• In the given problem, \( A = b \times 2b = 2b^2 \)

Understanding how to calculate the base area is crucial since it directly affects the volume. It provides the groundwork for obtaining the necessary components to use the pyramid volume formula effectively.

Volume Calculation

Calculating the volume of a pyramid is a step-by-step process that relies on understanding the formula and the dimensions involved. Here's how you can approach it:

• First, determine the base area of the pyramid.
• Then, identify the height of the pyramid.
• Substitute these values into the pyramid formula \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \).

For the problem at hand, substituting the base area as \( 2b^2 \) and height \( h \) into the formula gives:

• \( V = \frac{1}{3} \times 2b^2 \times h \).
• Simplifying this gives \( V = \frac{2}{3}b^2h \).

This demonstrates a typical method to solve for the volume of a pyramid when given the base dimensions and height. This comprehensive approach ensures a deep understanding of the relationships among the geometric properties of pyramids.