Problem 41 - Intermediate Algebra

Question

The volume of a cylinder varies jointly as its altitude and the square of the radius of its base. If a cylinder that has a base with a radius of 5 meters and has an altitude of 7 meters has a volume of \(549.5\) cubic meters, find the volume of a cylinder that has a base with a radius of 9 meters and has an altitude of 14 meters.

Step-by-Step Solution

Answer

The volume of the second cylinder is 3563.16 cubic meters.

Step 1: Understanding the Relationship

The problem states that the volume of a cylinder varies jointly as the altitude and the square of the radius of its base. Thus, we have the relationship: \( V = k \cdot h \cdot r^2 \), where \( V \) is the volume, \( h \) is the height, \( r \) is the radius, and \( k \) is a constant of proportionality.

Step 2: Finding the Constant of Proportionality

Given that a cylinder with a radius of 5 meters and a height of 7 meters has a volume of 549.5 cubic meters, we use the relationship to find \( k \). Substitute the values into the formula: \( 549.5 = k \cdot 7 \cdot 5^2 \). Simplify \(5^2\) to get 25, so \( 549.5 = k \cdot 7 \cdot 25 \). Solving \( k = \frac{549.5}{175} = 3.14 \).

Step 3: Calculating Volume for New Dimensions

Now, we use the same relationship to find the volume of a new cylinder with a radius of 9 meters and a height of 14 meters. Substitute the values into the formula with \( k = 3.14 \): \( V = 3.14 \cdot 14 \cdot 9^2 \). Simplify \(9^2\) to get 81. This gives \( V = 3.14 \cdot 14 \cdot 81 \).

Step 4: Final Calculation

Calculate \( V = 3.14 \cdot 14 \cdot 81 = 3563.16 \). Thus, the volume of the second cylinder is 3563.16 cubic meters.

Key Concepts

Joint Variation and Its Implications

Understanding the concept of joint variation is key to solving problems related to the volume of a cylinder. In mathematical terms, joint variation occurs when one variable is directly proportional to the product of two or more other variables. For instance, in the context of a cylinder's volume, it varies jointly as the product of its altitude (height) and the square of the base's radius. This can be expressed by the formula: \( V = k \cdot h \cdot r^2 \).

Here, \( V \) denotes the volume. The variables \( h \) and \( r \) represent the height and radius, respectively, while \( k \) acts as a constant which ensures the proper relation between the variables.

Joint variation is a fundamental concept that helps in describing real-world phenomena, allowing us to predict changes in one variable based on changes in the others. The relationship must be understood thoroughly to solve complex problems.

Understanding Proportionality Constant

The proportionality constant, often represented as \( k \), is a key component in equations that describe direct or joint variation. It essentially scales the relationship between the variables and is crucial for ensuring accuracy in calculations.

In the provided exercise, following the joint variation formula \( V = k \cdot h \cdot r^2 \), we solved for \( k \) using given dimensions of a cylinder — a radius of 5 meters and a height of 7 meters, which resulted in a volume of 549.5 cubic meters.

To find \( k \), substitute the known values into the formula and solve:

• Substitute: \( 549.5 = k \cdot 7 \cdot 5^2 \)
• Calculate \( 5^2 = 25 \)
• The equation becomes: \( 549.5 = k \cdot 7 \cdot 25 \)
• Solving for \( k \): \( k = \frac{549.5}{175} = 3.14 \)

The constant implies a specific scaling factor for the relationship, allowing the calculation of the volume for any other cylinder with given dimensions.

Algebraic Equations and Calculations

Algebraic equations provide a mechanism to describe relationships between different quantities mathematically. In this exercise, we utilized algebraic equations to determine the relationship between the volume of a cylinder, its radius, and its height.

We first formulated the problem using the equation \( V = k \cdot h \cdot r^2 \), which represents how the volume is calculated based on joint variation. By plugging in known values and solving algebraically, we determined the constant \( k \).

Once \( k \) was known, we used the same equation to solve for the volume of a different cylinder with a radius of 9 meters and a height of 14 meters. Here’s the step-by-step process:

• Substitute the known values: \( V = 3.14 \cdot 14 \cdot 9^2 \)
• Calculate \( 9^2 = 81 \)
• The equation becomes: \( V = 3.14 \cdot 14 \cdot 81 \)
• Finally, calculate: \( V = 3563.16 \)

This approach highlights how algebraic equations can simplify complex problems, providing a clear and methodical way to derive solutions by solving step by step.