Problem 64 - Algebra 2

Question

Physics In the expression \(P V^{\frac{7}{5}}, P\) represents the pressure and \(V\) represents the volume of a sample of a gas. Evaluate the expression for \(P=6\) and \(V=32\) .

Step-by-Step Solution

Answer

To find the value of the expression for \(P=6\) and \(V=32\), substitute these values into the given expression and simplify. First, calculate \(32^{\frac{7}{5}}\), and then multiply the resulting value by 6.

Step 1: Identify the Given Values

The values given in the problem are \(P=6\) (pressure) and \(V=32\) (volume).

Step 2: Substitute the Given Values into the Given Expression

The given expression is \(P V^{\frac{7}{5}}\). Substitute \(P=6\) and \(V=32\) into this expression to get: \(6 \cdot 32^{\frac{7}{5}}\).

Step 3: Perform the Operations to Simplify the Expression

First, calculate the value for \(32^{\frac{7}{5}}\), and then multiply it by 6. This will yield a precise value for the given expression.

Key Concepts

Pressure

Pressure is an important concept in physics, especially when dealing with gases. It is defined as the force exerted by gas particles against the walls of its container per unit area. Imagine small invisible particles colliding with the walls—they push against these walls, and this push is what we know as pressure. In the exercise, pressure is given by the symbol \( P \), and specifically it has a value of 6.
Understanding pressure is key for several applications:

• In weather systems, where it helps forecast storms and sunny days.
• In car tires, ensuring they travel safely and efficiently.
• In scuba diving, as pressure increases with depth underwater.

In mathematical terms, pressure is often involved in relationships and equations with other variables, such as volume, especially in gas law equations. It can be measured in units like pascals (Pa) or atmospheres (atm). To deepen your understanding, try to visualize how changing the pressure might affect the behavior of a confined gas.

Volume

Volume, in the context of gases, refers to the space that the gas occupies. Think of it as the container size in which the gas particles are bouncing around. In our exercise, the volume is represented by \( V \), with a specific value of 32.
Volume is an adaptable parameter, as gases can expand to fill their containers:

• In a balloon, the gas inside can inflate or deflate it.
• In baking, yeast releases gas to make dough rise.
• In air chambers, keeping the correct volume ensures proper function.

Measurement of volume is typically done in liters (L), cubic meters (m³), or milliliters (mL). A key idea is how volume changes can influence the properties of gases, like pressure. For example, decreasing the volume by compressing a gas often increases the pressure if the temperature remains unchanged. Play around with this idea using real-life examples, such as the change in volume of a bicycle pump when in use.

Gas laws

Gas laws are the principles that describe how gases behave under different conditions of pressure, volume, and temperature. These relationships enable us to predict how altering one factor will impact the others. The exercise involves these laws indirectly through the use of the expression \( P V^{\frac{7}{5}} \).
Some core gas laws include:

• Boyle's Law, which states that pressure is inversely proportional to volume when temperature is constant.
• Charles's Law, indicating that volume is directly proportional to temperature with constant pressure.
• Avogadro's Law, which tells us that volume is proportional to the number of gas particles at constant temperature and pressure.

The exponent \( \frac{7}{5} \) in our exercise plays into the realm of adiabatic processes, where there is no heat exchange with the environment. Understanding these fundamentals helps in solving complex problems and real-world applications, such as in engineering and environmental sciences. To build intuition, consider how these laws might apply to everyday phenomena like using a pressure cooker or the science behind a soda can fizzing when opened.