Math Snap

STEP 1

What is this asking?
If a gas's volume changes with temperature and pressure in a specific way, and we know its volume at one point, what's its volume at another point with a different temperature and pressure?
Watch out!
Don't mix up direct and inverse variation – they work in opposite ways!
Also, remember to round to the nearest integer at the end.

STEP 2

1. Set up the combined variation equation.
2. Find the constant of variation.
3. Calculate the new volume.

STEP 3

Alright, so we know the volume ($V$) varies directly with temperature ($T$) and inversely with pressure ($P$).
Direct variation means they change in the same direction – if one goes up, the other goes up!
Inverse variation means they change in opposite directions.

STEP 4

We can write this relationship as $V = k \cdot \frac{T}{P}$, where $k$ is our constant of variation.
This equation captures both the direct relationship with $T$ and the inverse relationship with $P$.

STEP 5

We're given an initial state: $V = \textbf{768} \mathrm{~m}^{3}$, $T = \textbf{268}$ K, and $P = \textbf{29}$ Pa.
Let's plug these values into our equation: $\textbf{768} = k \cdot \frac{\textbf{268}}{\textbf{29}}$.

STEP 6

To solve for $k$, we can multiply both sides by $\frac{\textbf{29}}{\textbf{268}}$.
This gives us $k = \textbf{768} \cdot \frac{\textbf{29}}{\textbf{268}}$.

STEP 7

Calculating this gives us $k \approx \textbf{83.02985}$.
We'll keep a few extra decimal places for accuracy and round at the very end.

STEP 8

Now we know $k$, and we're given a new temperature and pressure: $T = \textbf{325}$ K and $P = \textbf{11}$ Pa.
We can plug these, along with our calculated $k$, back into our equation: $V = \textbf{83.02985} \cdot \frac{\textbf{325}}{\textbf{11}}$.

STEP 9

Calculating this gives us $V \approx \textbf{2460.2985}$ $\mathrm{~m}^{3}$.

STEP 10

Finally, we need to round to the nearest integer, which gives us a final volume of approximately $\textbf{2460}$ $\mathrm{~m}^{3}$.

The volume of the gas at 325 K and 11 Pa is approximately 2460 $\mathrm{~m}^{3}$.