Problem 44 - College Algebra

Question

Use the four-step procedure for solving variation problems given on page 356 to solve. The volume of a gas varies directly as its temperature and inversely as its pressure. At a temperature of 100 Kelvin and a pressure of 15 kilograms per square meter, the gas occupies a volume of 20 cubic meters. Find the volume at a temperature of 150 Kelvin and a pressure of 30 kilograms per square meter.

Step-by-Step Solution

Answer

The volume of the gas at a temperature of 150 Kelvin and a pressure of 30 kilograms per square meter is 1500 cubic meters.

Step 1: Understand and write the relationship

From the problem, it's mentioned that the volume of the gas (let's calls it \(V\)) varies directly as its temperature (\(T\)) and inversely as its pressure (\(P\)). Write this relationship mathematically as \(V = kT / P\) where \(k\) is the constant of variation.

Step 2: Calculate the constant of variation

We know that at a temperature of 100 Kelvin and a pressure of 15 kg/square m, the volume of the gas is 20 cubic m. Substitute these values to find \(k\). \Solving \(20 = k(100) / 15\), gives \(k = 300\).

Step 3: Find the new volume

Now that we have \(k\), we can find the volume of gas at a temperature of 150 Kelvin and a pressure of 30 kg/square m. Substitute these constant and these values into \(V = kT / P\). \\(V = 300 (150) / 30 = 1500\).

Key Concepts

Direct Variation

Direct variation is a fundamental concept in mathematics that describes a relationship between two variables where one variable is a constant multiple of the other. In simpler terms, as one variable increases, the other variable increases at a consistent rate. This type of relationship can be represented by the equation \(y = kx\), where \(y\) and \(x\) are variables, and \(k\) is the constant of variation, also known as the proportionality constant.

In our exercise, the volume of the gas varies directly with temperature. This means that as the temperature increases, the volume also increases, provided that the pressure remains constant. Direct variation is essential for understanding how two factors are interrelated, helping us solve real-world problems involving proportional changes.

Inverse Variation

Inverse variation occurs when an increase in one variable results in a proportional decrease in another variable. In other words, the variables are inversely proportional to each other. This relationship is traditionally expressed by the formula \(xy = k\) or \(y = \frac{k}{x}\), where \(k\) is a non-zero constant.

For the given problem, the volume of a gas varies inversely with pressure. This indicates that if the pressure increases while the temperature remains constant, the volume decreases. Inverse variation is a key concept in grasping how two variables react when one undergoes a change, commonly seen in scenarios like physics and economics.

Constant of Variation

The constant of variation is a crucial part of understanding both direct and inverse variation relationships. It represents the fixed value that relates the two variables. The constant of variation remains unchanged, serving as the scale factor in direct variation and the product or quotient in inverse variation.

In our example, we determined the constant of variation \(k\) to be 300. This constant ties the volume of the gas to its temperature and pressure, allowing us to predict the volume under different conditions. It is derived from known values of the variables and is pivotal in forming the direct and inverse variation equations.

Temperature

Temperature is a measure of heat energy within a system and is typically measured in degrees Celsius, Fahrenheit, or Kelvin. It plays a vital role in various scientific calculations and real-world applications.

In the context of gas laws, temperature is directly related to the volume of a gas. Higher temperatures cause particles to move faster, leading to an increased volume. This relationship is captured in our problem, where the volume of gas changes as the temperature varies. Understanding temperature's role in variation problems helps illuminate how thermal changes impact physical properties.

Pressure

Pressure is the force exerted per unit area and is usually expressed in units like pascals, atmospheres, or kilograms per square meter. It is an essential factor in many physical phenomena and engineering applications.

According to our problem, the volume of the gas varies inversely with pressure. This means that as pressure on the gas increases, its volume decreases if temperature is held constant. This concept is integral to the study of gases and helps explain everyday occurrences, such as why balloons expand in lower pressure environments. Understanding pressure in the context of inverse variation highlights its critical role in affecting volumes in closed systems.