Problem 12 - Thomas Calculus

Question

Find the volume of the given pyramid, which has a square base of area 9 and height \(5 .\)

Step-by-Step Solution

Answer

The volume of the pyramid is 15 cubic units.

Step 1: Identify the Formula

To find the volume of a pyramid, use the formula \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \).

Step 2: Substitute the Given Values

Plug in the values given in the problem: \( \text{Base Area} = 9 \) and \( \text{Height} = 5 \) into the formula. This gives us \( V = \frac{1}{3} \times 9 \times 5 \).

Step 3: Calculate the Volume

Calculate the expression: \( V = \frac{1}{3} \times 9 \times 5 = 15 \).

Key Concepts

Geometric Formulas

Understanding geometric formulas is crucial in solving volume problems. Specifically, the formula for the volume of a pyramid is quite straightforward. It is given as:

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

This formula shows that the volume is one-third the product of the base area and the height of the pyramid. Breaking down the components:

• Base Area: The base can be any shape, but in this problem, it's a square.
• Height: This is the perpendicular distance from the base to the apex.

It's essential to remember that the base area and height must be in the same unit of measure to use this formula effectively.
Consistently applying the correct formula will lead to the right solution.

Mathematical Problem Solving

Mathematical problem solving often involves applying known formulas to insert the given values, simplifying calculations. In this exercise, our goal is to find the volume of a pyramid.
The process involves:

• Identifying key elements: Recognize that the problem deals with a pyramid with a square base.
• Plugging in values: Use the values provided for the base area and height. Here, base area = 9 and height = 5.
• Performing calculations: Calculate the volume using the formula, resulting in \( V = \frac{1}{3} \times 9 \times 5 = 15 \).

These steps ensure a clear path from problem statement to solution. Double-check calculations to ensure accuracy.

Pyramid Properties

Pyramids are widely studied in geometry due to their unique properties.

• They have a polygon as a base—the shape of which defines the pyramid type.
• All faces other than the base are triangular and converge at a single point called the apex.

In this problem, the base is a square, the most common and simplest pyramid form. Knowing the properties helps in visualizing how the base area and height affect the volume.
The height, also known as the vertical height, is particularly important because it is always perpendicular to the base. It's an essential dimension for calculating volume accurately. Understanding these properties helps in both theoretical and practical applications of pyramids in geometry.