Problem 60 - Algebra 2

Question

Describe the variation that is modeled by each formula. \(V=\frac{s^{2} h}{3}\)

Step-by-Step Solution

Answer

The formula \(V=\frac{s^{2} h}{3}\) describes a direct square variation with a constant of variation of \(\frac{1}{3}\). The volume \(V\) of an object (like a pyramid or cone) varies directly as the square of the side length (\(s^2\)) and the height (\(h\)), reduced by a factor of \(\frac{1}{3}\).

Step 1: Identify the Variables

The formula \(V=\frac{s^{2} h}{3}\) involves three variables, where \(V\) stands for volume, \(s\) is the side length of the base, and \(h\) is the height. This formula is often used for calculating the volume of a pyramid or a cone.

Step 2: Explain the Variation

In this formula, if you keep the height \(h\) constant, the volume \(V\) varies directly with the square of the side length, which is \(s^2\). This is known as a direct square variation. If \(s\) is doubled, the volume \(V\) will quadruple; if \(s\) is halved, the volume \(V\) will be quartered, and so on.

Step 3: Explain the Constant of Variation

The constant of variation here is \(\frac{1}{3}\), which is constant because it does not vary. This means that the volume \(V\) is one-third of the product of the square of the side length and the height. This factor originates from the geometric property of a pyramid. This \(\frac{1}{3}\) makes the volume of a pyramid or cone three times smaller than the volume of a prism or cylinder with the same base and height.

Key Concepts

Direct Square Variation

In mathematical functions, a **direct square variation** is when a variable varies directly as the square of another variable. In simpler terms, this means one quantity changes in direct proportion to the square of another. In the context of the formula \(V=\frac{s^{2} h}{3}\), the volume \(V\) changes as the square of the side length \(s\) of the pyramid or cone's base changes.
Keeping the height \(h\) constant, any increase in the side length \(s\) will result in a significant change in the volume. This relationship can be observed by:

• If the side length \(s\) is doubled, the volume increases by a factor of four (since \(2^2 = 4\)).
• If the side length \(s\) is halved, the volume decreases by a factor of four.

Understanding this variation helps predict how the volume changes with different side lengths, which is crucial in practical applications such as architecture and design.

Geometric Properties

Geometric properties play a significant role in understanding the structures of shapes, like pyramids and cones. These properties help us derive formulas to calculate dimensions such as volume. A pyramid or cone has a specific attribute—its volume is distinct compared to a prism or a cylinder.
The formula \(V=\frac{s^{2} h}{3}\) showcases this attribute. It describes how the dimensions determine the space within the shape.
In this formula:

• The area of the base is represented by \(s^2\), which is the square of the side length. This area is a crucial geometric property, as it influences the overall volume.
• The height \(h\) determines how tall the shape is, akin to stretching the base upwards.

Understanding these geometric properties is essential as they explain why specific formulas don't just arise from arbitrary calculations, but from intrinsic properties of the shapes themselves.

Constant of Variation

The **constant of variation** is a coefficient that assists in describing how one quantity changes in relation to another. In our formula \(V=\frac{s^{2} h}{3}\), the constant is \(\frac{1}{3}\). This constant is vital as it adjusts the relationship between the square of the side length \(s\) and the height \(h\) to correctly represent the volume of a pyramid or cone.
Without this constant, the formula would resemble that of a prism or cylinder. Here’s why the constant of \(\frac{1}{3}\) is significant:

• It reduces the product of the base area and height to accurately find the volume of the pyramid.
• This factor accounts for the tapering shape of the pyramid, which means it's less voluminous than a prism with an identical base and height.

Grasping the concept of a constant of variation provides a deeper understanding of why geometric formulas differ for pyramids and prisms, reflecting their unique spatial properties.