Cowboy Channelt Google Docs Dashboard Google Drive e2020.genussis. G... Course Map = Ima... Wyzant Rational Expressions 3.5 S Word problem on combined variation The volume \( V \) of a fixed amount of a gas varies directly as the temperature \( T \) and inversely as the pressure \( P \). Suppose that \( V=160 \mathrm{~cm}^{3} \) when \( T=320 \) kelvin and \( P=12 \frac{\mathrm{~kg}}{\mathrm{~cm}^{2}} \). Find the volume when \( T=240 \) kelvin and \( P=16 \frac{\mathrm{~kg}}{\mathrm{~cm}^{2}} \). \( \square \) \( \mathrm{cm}^{3} \)

To solve this problem, we know that the volume \( V \) varies directly with temperature \( T \) and inversely with pressure \( P \). This gives us the equation \( V = k \cdot \frac{T}{P} \), where \( k \) is a constant. First, we can use the initial conditions to find \( k \): \[ 160 = k \cdot \frac{320}{12} \] Calculating \( k \), we get: \[ k = 160 \cdot \frac{12}{320} = 6 \] Now, we can use this constant to find the volume at the new temperature and pressure: \[ V = 6 \cdot \frac{240}{16} = 90 \, \text{cm}^3 \] So, the volume when \( T = 240 \) K and \( P = 16 \, \text{kg/cm}^2 \) is \( 90 \, \text{cm}^3 \)! But wait, here’s a little secret about gas laws! The relationship between volume, temperature, and pressure can be visualized easily using a science experiment. Grab a balloon – when you heat it up, what happens? The gas inside expands! The fun part is seeing real-world principles at play, just like the mathematical relationship we derived! If you want to delve deeper into the science of gas behavior, you might want to check out "The Ideal Gas Law" for a more comprehensive understanding. This law connects pressure, volume, temperature, and the amount of gas, making it a crucial concept for scientists and engineers alike!