The problem states that for every 2 psi drop in pressure, the temperature rises by 3 degrees. This indicates a linear relationship between temperature and pressure, T = mP + c.
This physics problem tests your understanding of fundamental physical laws and their applications. The step-by-step solution below breaks down the problem using relevant equations and physical reasoning.
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Question 3
1. Derive the formula which shows the interrelationship between pressure and temperature.
Let $P$ be the pressure in psi and $T$ be the temperature in degrees.
The problem states that for every 2 psi drop in pressure, the temperature rises by 3 degrees. This indicates a linear relationship between temperature and pressure, $T = mP + c$.
Step 1: Determine the slope ($m$).
The change in pressure is $\Delta P = -2$ psi (pressure drops).
The change in temperature is $\Delta T = +3$ degrees (temperature rises).
The slope $m$ is given by:
$$m = \frac{\Delta T}{\Delta P} = \frac{3}{-2} = -\frac{3}{2}$$
Step 2: Use the given point to find the y-intercept ($c$).
We are given that when the pressure is $P = 2$ psi, the temperature is $T = -12$ degrees.
Substitute these values and the slope into the linear equation $T = mP + c$:
$$-12 = \left(-\frac{3}{2}\right)(2) + c$$
$$-12 = -3 + c$$
Step 3: Solve for $c$.
$$c = -12 + 3$$
$$c = -9$$
Step 4: Write the final formula.
Substitute the slope $m = -\frac{3}{2}$ and the y-intercept $c = -9$ into $T = mP + c$:
$$T = -\frac{3}{2}P - 9$$
The formula showing the interrelationship between pressure and temperature is:
$$\boxed{T = -\frac{3}{2}P - 9}$$
2. Draw the graph of the above function, show the intersection points on the graph with the axes precisely.
The function is $T = -\frac{3}{2}P - 9$.
To draw the graph, we find the intercepts with the axes.
Step 1: Find the T-intercept (where $P=0$).
Substitute $P=0$ into the equation:
$$T = -\frac{3}{2}(0) - 9$$
$$T = -9$$
The T-intercept is $(0, -9)$.
Step 2: Find the P-intercept (where $T=0$).
Substitute $T=0$ into the equation:
$$0 = -\frac{3}{2}P - 9$$
Add 9 to both sides:
$$9 = -\frac{3}{2}P$$
Multiply by $-\frac{2}{3}$ to solve for $P$:
$$P = 9 \times \left(-\frac{2}{3}\right)$$
$$P = -3 \times 2$$
$$P = -6$$
The P-intercept is $(-6, 0)$.
Step 3: Draw the graph.
Plot the two intercepts $(0, -9)$ and $(-6, 0)$ and draw a straight line through them.
The horizontal axis represents Pressure ($P$ in psi) and the vertical axis represents Temperature ($T$ in degrees).
`
^ T (degrees)
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