A heater of resistance 20 `Omega` is used to heat a room at `10^@C` and is connected to 220 V mains. The temperature is uniform inside the room and heat is transmitted outside the room through a glass window of area `1.2 m^2` and thickness 0.1 cm. If `T^@C` is the temperature outside and thermal conductivity of glass is` 0.2 cal s^(-1) m^(-1) ""^@C^(-1)` , then find the value of 10T - 75 close to nearest integer.

Explanation

To solve the problem step by step, we will follow the process of calculating the power generated by the heater, the heat loss through the glass window, and finally find the temperature outside (T) and the required value of \(10T - 75\).

### Step 1: Calculate the power generated by the heater
The power \(P\) generated by the heater can be calculated using the formula:
\[ P = \frac{V^2}{R} \]
where \(V\) is the voltage (220 V) and \(R\) is the resistance (20 Ω).

\[ P = \frac{(220)^2}{20} = \frac{48400}{20} = 2420 \text{ W} \]

### Step 2: Convert power from watts to calories per second
To convert watts to calories per second, we use the conversion factor \(1 \text{ W} = \frac{1}{4.2} \text{ cal/s}\).

\[ P = \frac{2420}{4.2} \approx 575.6 \text{ cal/s} \]

### Step 3: Use the formula for heat transfer through the glass window
The rate of heat transfer through the glass window can be calculated using Fourier's law of heat conduction:
\[ \frac{Q}{t} = \frac{k \cdot A \cdot (T_2 - T_1)}{L} \]
where:
- \(Q/t\) is the rate of heat transfer (cal/s),
- \(k\) is the thermal conductivity of glass (0.2 cal/s·m·°C),
- \(A\) is the area of the window (1.2 m²),
- \(T_2\) is the inside temperature (10 °C),
- \(T_1\) is the outside temperature (T °C),
- \(L\) is the thickness of the glass (0.1 cm = 0.001 m).

Substituting the values, we have:
\[ 575.6 = \frac{0.2 \cdot 1.2 \cdot (10 - T)}{0.001} \]

### Step 4: Simplify the equation
Rearranging the equation gives:
\[ 575.6 = \frac{0.24 \cdot (10 - T)}{0.001} \]
\[ 575.6 = 240 \cdot (10 - T) \]

### Step 5: Solve for \(T\)
Now, divide both sides by 240:
\[ \frac{575.6}{240} = 10 - T \]
\[ 2.3983 \approx 10 - T \]
\[ T \approx 10 - 2.3983 \approx 7.6017 \]

### Step 6: Calculate \(10T - 75\)
Now, we need to find \(10T - 75\):
\[ 10T - 75 = 10 \cdot 7.6017 - 75 \]
\[ = 76.017 - 75 \approx 1.017 \]

Rounding to the nearest integer gives:
\[ \text{Final Answer} \approx 1 \]