To determine the outside temperature based on the given conditions, we need to analyze the heat transfer occurring through the glass window and the heat generated by the resistance heater in the room. Let's break this down step by step.
Step 1: Calculate the Heat Generated by the Resistance Heater
The power (P) generated by the resistance heater can be calculated using Ohm's law, which states that power is equal to voltage squared divided by resistance:
P = V² / R
Substituting the given values:
- V = 200 volts
- R = 20 ohms
P = (200)² / 20 = 40000 / 20 = 2000 watts
Step 2: Convert Power to Calories
Since we need to work with calories for the thermal conductivity, we convert watts to calories. The conversion factor is:
1 watt = 0.239006 calories/second
Thus, the power in calories is:
Power in calories = 2000 watts × 0.239006 cal/watt = 478.012 calories/second
Step 3: Calculate the Heat Loss through the Glass Window
The heat loss through the window can be calculated using the formula for conduction:
Q = k × A × (T_inside - T_outside) / d
Where:
- Q = heat transfer (calories/second)
- k = thermal conductivity (0.2 cal/m·°C·sec)
- A = area of the window (1 m²)
- T_inside = inside temperature (20°C)
- T_outside = outside temperature (unknown)
- d = thickness of the glass (0.2 cm = 0.002 m)
Substituting the known values into the equation:
478.012 = 0.2 × 1 × (20 - T_outside) / 0.002
Step 4: Simplifying the Equation
Now, we can simplify the equation:
478.012 = 100 × (20 - T_outside)
Dividing both sides by 100:
4.78012 = 20 - T_outside
Step 5: Solve for T_outside
Rearranging the equation gives:
T_outside = 20 - 4.78012
T_outside = 15.21988°C
Rounding to two decimal places, we find:
T_outside ≈ 15.22°C
Final Result
Thus, the calculated outside temperature is approximately 15.22°C. This result aligns closely with the expected answer of 15.24°C, considering rounding and approximation differences. Understanding these calculations helps illustrate the principles of heat transfer and thermal dynamics in practical scenarios.
