Problem 9 - Intermediate Algebra

Question

Translate each statement of variation into an equation, and use \(k\) as the constant of variation. The volume \((V)\) of a cone varies jointly as its height and the square of its radius.

Step-by-Step Solution

Answer

The equation is \(V = k \, h \, r^2\).

Step 1: Understanding the Problem

The problem states that the volume of a cone varies jointly with height and the square of the radius. This means that the volume is directly proportional to both the height and the square of the radius.

Step 2: Identifying the Variables

We have three variables in this situation: the volume of the cone, which is denoted as \(V\), the height of the cone, denoted as \(h\), and the radius of the cone, denoted as \(r\). These variables are related through the equation we need to find.

Step 3: Setting Up the Equation

Since the volume of the cone varies jointly as the height and the square of the radius, we can write this relationship as: \(V = k \, h \, r^2\), where \(k\) is the constant of variation.

Step 4: Final Equation

The final equation expressing the statement of variation is \(V = k \, h \, r^2\). This represents the joint variation of the volume with the height and the square of the radius.

Key Concepts

Joint Variation

In mathematical terms, joint variation involves a situation where one variable depends on two or more other variables. It is like a combination of direct proportionality scenarios, where different variables influence each other simultaneously. In this specific case of the volume of a cone, the volume

• varies jointly as the height,
• and also as the square of the radius.

To express joint variation, we use an equation that multiplies each influencing variable along with a constant of variation, typically denoted as \(k\). Applying this concept, the formula for the volume \(V\) of the cone, in terms of joint variation, becomes:

• \(V = k \, h \, r^2\)

Here, \(h\) represents the height of the cone and \(r\) represents the radius. Understanding that the variables are not isolated but rather work in conjunction will make complex relationships in geometry much more approachable.

Direct Proportionality

Direct proportionality occurs when two quantities increase or decrease at the same rate. In the context of how it's relevant to our exercise, the volume of a cone is directly proportional to both its height and its radius squared. This means that:

• If the height of the cone doubles and the radius remains the same, the volume doubles as well.
• Similarly, if the square of the radius doubles and the height stays the same, the volume will also double.

The constant of variation \(k\) does not change, and it characterizes the specific relationship between these variables. This concept allows us to predict how changes in the cone's dimensions (height and radius) will affect its overall volume, maintaining a consistent proportional relationship ever-changing.

Volume of a Cone

The volume of a cone is a measure of the amount of space inside it. It's one of the most important characteristics of cones in geometry. The formula for calculating the volume \(V\) of a cone is represented by the formula:

• \(V = \frac{1}{3} \pi r^2 h\)

Here, \(\pi\) is a constant approximately equal to 3.14159, \(r\) is the radius of the cone's base, and \(h\) is the height of the cone. However, in specific problems involving variation, this formula transforms into a joint variation equation such as:

• \(V = k \, h \, r^2\)

This emphasizes the relationship that \(V\) changes as \(h\) and \(r^2\) change. Understanding this formula helps make sense of how dimensions are directly tied to the space within a cone, a concept crucial not only in mathematics but also in real-world applications like engineering and architecture.