A wall of dimensions 2.00 m by `3.50 m` has a single-pane window of dimensions `0.75 m` by `1.20 m` . If the inside temperature is `20^(@)C` and the outside temperature is `-10^(@)C` , effective thermal resistance of the opaque wall and window are `2.10 K W^(-1)` and `0.21 K W^(-1)` respectively. The heat flow through the entire wall will be .

Options

A. `215 W`
B. `205 W`
C. `175 W`
D. `110 W`

Explanation

To solve the problem of calculating the heat flow through the wall and the window, we will follow these steps:

### Step 1: Calculate the area of the window
The dimensions of the window are given as \(0.75 \, m\) by \(1.20 \, m\).

\[ \text{Area of the window} = \text{width} \times \text{height} = 0.75 \, m \times 1.20 \, m = 0.90 \, m^2 \]

### Step 2: Calculate the area of the wall
The dimensions of the wall are given as \(2.00 \, m\) by \(3.50 \, m\).

\[ \text{Area of the wall} = \text{width} \times \text{height} = 2.00 \, m \times 3.50 \, m = 7.00 \, m^2 \]

### Step 3: Calculate the area of the opaque part of the wall
To find the area of the opaque part of the wall, we subtract the area of the window from the area of the wall.

\[ \text{Area of opaque wall} = \text{Area of wall} - \text{Area of window} = 7.00 \, m^2 - 0.90 \, m^2 = 6.10 \, m^2 \]

### Step 4: Calculate the temperature difference
The inside temperature is \(20^\circ C\) and the outside temperature is \(-10^\circ C\).

\[ \text{Temperature difference} = T_{\text{inside}} - T_{\text{outside}} = 20 - (-10) = 30 \, K \]

### Step 5: Calculate the heat flow through the window
Using the formula for heat flow \( \frac{Q}{t} = \frac{\Delta T}{R} \), where \(R\) is the thermal resistance:

\[ \text{Heat flow through window} = \frac{\text{Area of window} \times \text{Temperature difference}}{\text{Thermal resistance of window}} \]

Substituting the values:

\[ \text{Heat flow through window} = \frac{0.90 \, m^2 \times 30 \, K}{0.21 \, K \cdot W^{-1}} = \frac{27 \, m^2 \cdot K}{0.21 \, K \cdot W^{-1}} \approx 128.57 \, W \]

### Step 6: Calculate the heat flow through the opaque wall
Using the same formula for the opaque wall:

\[ \text{Heat flow through opaque wall} = \frac{\text{Area of opaque wall} \times \text{Temperature difference}}{\text{Thermal resistance of opaque wall}} \]

Substituting the values:

\[ \text{Heat flow through opaque wall} = \frac{6.10 \, m^2 \times 30 \, K}{2.10 \, K \cdot W^{-1}} = \frac{183 \, m^2 \cdot K}{2.10 \, K \cdot W^{-1}} \approx 87.14 \, W \]

### Step 7: Calculate the total heat flow
Now, we add the heat flow through the window and the opaque wall to find the total heat flow:

\[ \text{Total heat flow} = \text{Heat flow through window} + \text{Heat flow through opaque wall} \]

Substituting the values:

\[ \text{Total heat flow} = 128.57 \, W + 87.14 \, W \approx 215.71 \, W \]

### Final Answer
The total heat flow through the entire wall is approximately \(216 \, W\).

---