Problem 41 - Elementary and Intermediate Algebra

Question

Solve each problem by writing a variation model. Under constant temperature, the volume occupied by a gas varies inversely to the pressure applied. If the gas occupies a volume of 20 cubic inches under a pressure of 6 pounds per square inch, find the volume when the gas is subjected to a pressure of 10 pounds per square inch.

Step-by-Step Solution

Answer

The volume of the gas is 12 cubic inches when the pressure is 10 pounds per square inch.

Step 1: Understand Variation Model

The problem describes an inverse variation: the volume of gas, \( V \), varies inversely with the pressure, \( P \). This relationship can be modeled by the equation \( V = \frac{k}{P} \), where \( k \) is a constant.

Step 2: Determine the Constant of Variation

We know from the problem that the gas occupies 20 cubic inches under a pressure of 6 pounds per square inch. Using \( V = \frac{k}{P} \), substitute \( V = 20 \) and \( P = 6 \) to find \( k \):\[ 20 = \frac{k}{6} \]Solving for \( k \):\[ k = 20 imes 6 = 120 \]

Step 3: Apply Variation Model to New Pressure

Now use the variation model to find the volume when the pressure is 10 pounds per square inch. Substitute \( k = 120 \) and \( P = 10 \) into the equation:\[ V = \frac{120}{10} \]Calculate \( V \):\[ V = 12 \]

Step 4: Conclude with Solution

Thus, when the pressure is increased to 10 pounds per square inch, the volume of the gas is reduced to 12 cubic inches.

Key Concepts

Volume and Pressure Relationship

In the study of gases, the relationship between volume and pressure is a classic illustration of inverse variation. When we talk about these two parameters, we're essentially discussing how volume changes as pressure changes, with one affecting the other in opposite directions. Imagine a balloon—when you squeeze it, its volume decreases, and the pressure increases. Essentially, for a given amount of gas at a constant temperature, as you increase the pressure exerted on the gas, the volume decreases, and vice versa. This concept is encapsulated in Boyle's Law, which tells us that the product of volume and pressure is a constant if temperature remains unchanged. Therefore, we understand that volume varies inversely with pressure: as one goes up, the other comes down. This kind of interaction is fundamental in understanding how gases behave under different conditions.

Variation Models

Variation models are mathematical equations used to describe how one quantity changes in relation to another. In the context of volume and pressure of gases, we use an inverse variation model. This model is represented by the equation: \[ V = \frac{k}{P} \]where:

• \( V \) = Volume of the gas
• \( P \) = Pressure applied
• \( k \) = Constant of variation

Through this equation, we determine how the pressure affects the volume and vice versa. The inverse relationship means increasing pressure results in decreased volume and is crucial for calculations involving gases, whether it's within an engine, a balloon, or even weather balloons used in meteorology. Once you master setting up these models, it becomes much easier to predict the behavior of gases.

Constant of Variation

The constant of variation, symbolized as \( k \), is key to the inverse variation equation. It remains consistent for a given condition under the same temperature. It essentially quantifies the relationship between the variables involved. Using the example from our exercise, we initially know that the volume is 20 cubic inches when the pressure is 6 pounds per square inch. This gives us the equation: \[ 20 = \frac{k}{6} \]Solving for \( k \), which represents the pressure-volume interaction's constant value under specific conditions, we get \( k = 120 \). This constant allows us to predict new scenarios, such as the new volume when pressure changes. Its discovery enables consistent calculations across different situations, ensuring that the mathematical relationship holds steady as long as the temperature remains unchanged. The understanding of \( k \) is central because it bridges direct numerical representation with real-world physical phenomena.