1. Given data

• Furnace temperature: $T=360\,\text{K}$
• Atmospheric temperature: $T_a=300\,\text{K}$
• Atmospheric density: $\rho_a=1.2\,\text{kg m}^{-3}$
• Furnace height: $H=1\,\text{m}$
• Chimney height above furnace: $h=9\,\text{m}$
• Thus total height of hot air column from furnace bottom to chimney top: $H+h=1+9=10\,\text{m}$
• Atmospheric pressure at furnace inlet and outside air is $P_a$.

We need the pressure difference across the cap when chimney top is closed.

2. Density of hot air inside furnace/chimney

Since air behaves as an ideal gas and heating is at constant pressure, $\rho \propto \frac{1}{T}$ So, $\rho = \rho_a\frac{T_a}{T}$ $\rho = 1.2\times \frac{300}{360} = 1.2\times \frac{5}{6} = 1.0\,\text{kg m}^{-3}$

Thus hot air density is $\rho=1.0\,\text{kg m}^{-3}$

3. Pressure at the bottom of furnace

The furnace is open to atmosphere at the bottom inlet, so pressure there is atmospheric: $P_{\text{bottom}}=P_a$

Inside the closed chimney-furnace system, hot air of density $\rho$ fills the column of height $H+h=10\,\text{m}$. So pressure decreases upward hydrostatically by $\rho g(H+h)$

Hence pressure just below the cap (inside chimney top) is $P_{\text{in}}=P_a-\rho g(H+h)$

Outside the cap, the pressure is the atmospheric pressure at the same height. Starting from pressure $P_a$ at the furnace bottom level, and moving up through outside atmospheric air of density $\rho_a$ by height $10\,\text{m}$, $P_{\text{out}}=P_a-\rho_a g(H+h)$

4. Pressure difference across cap

The pressure difference between bottom and top surfaces of the cap is $\Delta P=P_{\text{out}}-P_{\text{in}}$

Substitute: $\Delta P=\left(P_a-\rho_a g(H+h)\right)-\left(P_a-\rho g(H+h)\right)$ $\Delta P=(\rho-\rho_a)g(H+h)$ In magnitude, $\Delta P=(\rho_a-\rho)g(H+h)$

Now, $\rho_a-\rho = 1.2-1.0=0.2\,\text{kg m}^{-3}$ Therefore, $\Delta P=0.2\times 10\times 10 = 20\,\text{N m}^{-2}$

5. Final answer

$\boxed{20\,\text{N m}^{-2}}$

6. Comparison with stored answer

Stored correct answer = $30$

My derived answer is $20$, so I do not agree with the stored answer.

The likely reason is that the pressure difference should be computed over the full hot-air column height from furnace bottom to chimney top, which is $H+h=10\,\text{m}$, and using ideal-gas density relation gives $\rho=1.0\,\text{kg m}^{-3}$. This yields $20\,\text{N m}^{-2}$, not $30\,\text{N m}^{-2}$.