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 is 15 cubic units.

Step 1: Understand the Formula for Volume of a Pyramid

The volume \( V \) 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 2: Identify the Given Values

We are given that the area of the base \( B = 9 \) and the height \( h = 5 \). These values will be substituted into the volume formula.

Step 3: Plug Values into the Formula

Substitute \( B = 9 \) and \( h = 5 \) into the volume formula: \( V = \frac{1}{3} imes 9 imes 5 \).

Step 4: Perform the Arithmetic

Calculate the volume: \( V = \frac{1}{3} imes 45 = 15 \).

Step 5: State the Final Answer

The volume of the pyramid is 15 cubic units.

Key Concepts

Pyramid Formula

To find the volume of a pyramid, you need to use a specific mathematical formula. The formula is: \( V = \frac{1}{3} B h \). Here, \( V \) stands for the volume, \( B \) represents the area of the base, and \( h \) denotes the height of the pyramid.
This formula is derived from the fact that a pyramid is essentially one-third of a prism with the same base area and height.

• This division by three accounts for the pyramid's tapering shape, which reduces its volume compared to a prism.
• This formula works for any pyramid, regardless of whether it has a triangular, square, or any other polygonal base.

Thus, understanding and applying this formula correctly is crucial for solving problems related to the volume of pyramids.

Area of Base

The area of the base of a pyramid is a key factor in determining its volume. For any pyramid, the base can take various shapes such as triangles, squares, or any other polygons.
In this specific exercise, the base is a square, and its area is already given as 9 square units.

• For squares, the area is calculated by squaring the side length: \( s^2 \).
• If the base were another shape, you would need to use the appropriate formula for that shape's area (e.g., for rectangles, it's length \( \times \) width).

The accurate calculation of the base area is essential because errors here will directly affect the final result of the pyramid's volume. Make sure to always use the correct formula depending on the base's shape.

Height of Pyramid

The height of a pyramid, symbolized as \( h \), is the perpendicular distance from its apex (top point) to the center of the base. This measurement is crucial in calculating the pyramid's volume.
In this exercise, the given height of the pyramid is 5 units.

• Always ensure the height is measured perpendicular to the base, not slant height or any side length of the pyramid.
• Confusing the height with other measures can lead to incorrect calculations of the volume.

Understanding what measures are needed and how they fit into formulas helps in mathematically visualizing and solving pyramid-related problems.