Problem 33 - Intermediate Algebra

Question

The volume \((V)\) of a gas varies directly as the temperature \((T)\) and inversely as the pressure \((P)\). If \(V=\) 48 when \(T=320\) and \(P=20\), find \(V\) when \(T=280\) and \(P=30\).

Step-by-Step Solution

Answer

The volume \( V \) is 28 when \( T = 280 \) and \( P = 30 \).

Step 1: Understand the Relationship

The volume \( V \) of a gas is related to the temperature \( T \) and the pressure \( P \) by a direct and inverse variation. This can be expressed as \( V = k \frac{T}{P} \), where \( k \) is a constant.

Step 2: Determine the Constant \( k \)

Using the given values, substitute \( V = 48 \), \( T = 320 \), and \( P = 20 \) into the equation \( V = k \frac{T}{P} \), to find \( k \). Substitute to get:\[48 = k \frac{320}{20}\]Solve for \( k \):\[48 = k \cdot 16\]\[k = \frac{48}{16} = 3\]

Step 3: Use the Constant to Find New Volume

With \( k = 3 \), find the new volume \( V \) when \( T = 280 \) and \( P = 30 \). Use the formula:\[V = 3 \cdot \frac{280}{30}\]Simplify:\[V = 3 \cdot \frac{28}{3}\]\[V = 28\]

Step 4: Verify the Result

Ensure the calculations are correct and consistent with the derived formula and the relationship defined. The new volume at the given conditions is correctly calculated.

Key Concepts

Direct Variation

Direct variation refers to a relationship between two variables where they increase or decrease together at a constant rate. This type of relationship is represented by the equation \( y = kx \), where \( y \) is directly proportional to \( x \) and \( k \) is the constant of proportionality. A simple analogy is a car traveling at a constant speed: the distance traveled increases linearly with time.In the context of gas laws, the volume of a gas (\( V \)) and its temperature (\( T \)) are directly related, assuming pressure remains constant. This means that as temperature increases, volume also increases, and vice versa.

• If you double the temperature, you double the volume.
• The constant \( k \) is essential, as it connects \( V \) and \( T \) in this relationship.

Understanding direct variation helps predict how changing one variable affects the other in real-life scenarios.

Inverse Variation

Inverse variation occurs when one variable increases while the other decreases, maintaining a constant product. This is expressed mathematically as \( xy = k \), or equivalently, \( y = \frac{k}{x} \). Imagine two people sharing a fixed amount of candy. As one person's share increases, the other's share decreases proportionally.Within gas laws, inverse variation describes the relationship between volume (\( V \)) and pressure (\( P \)) of a gas at a constant temperature. Thus, if the pressure increases, the volume must decrease to keep their product constant, and vice versa.

• If you halve the pressure, the volume doubles.
• This inverse relationship signifies conservation of energy and matter under changing conditions.

The inverse variation principle is crucial for understanding behaviors such as how gases compress under higher pressures.

Proportional Relationships

Proportional relationships describe how two quantities change in relation to one another at a constant rate. This can be either a direct or an inverse variation. Such relationships are foundational in many scientific laws, including those governing gases.In the exercise provided, the problem showcases both direct and inverse variations through the combined gas law. The volume of a gas is directly proportional to its temperature and inversely proportional to its pressure. This relationship is captured in the formula \( V = k \frac{T}{P} \).

• Direct proportion: Increasing temperature leads to an increase in volume.
• Inverse proportion: Increasing pressure leads to a decrease in volume.

By understanding these proportional relationships, you can predict how changing one variable affects others, allowing for practical applications in fields like chemistry and physics.