**Step 1: Conceptual Introduction** This problem deals with the heat loss from a window during wintertime, where the window is poorly insulated. The main mode of heat transfer is convection, as the insulation reduces radiative losses. Heat is lost from the warm room to the cold window through a gap between the insulation and the window, and we need to find both the rate of heat loss (in watts) and the weekly energy cost due to this loss. **Explanation block:** We are given the dimensions of the window, the temperatures on both sides, and the thickness of the air gap (where convection occurs). The ill-fitting insulation means air can move freely in this gap, so heat will be lost primarily through convection. We’ll use the temperature difference and the properties of the air gap to compute heat loss. --- **Step 2: Formulae Used** The heat loss by convection through the air gap is given by: \[ Q = \frac{k \cdot A \cdot \Delta T}{d} \] Where: - \( Q \) = Rate of heat loss (W) - \( k \) = Thermal conductivity of air (\( \approx 0.026 \) W/m·K) - \( A \) = Area of window (m²) - \( \Delta T \) = Temperature difference (K or °C) - \( d \) = Thickness of air gap (m) To find cost: - Weekly energy loss = \( Q \times \text{time in seconds per week} \) - Convert energy to kWh, then multiply by electricity rate. **Explanation block:** The formula above calculates the convective heat transfer through the air gap, treating the air as a stationary layer. Since the insulation mainly blocks radiation, convection dominates. The area is the surface area of the window, and the temperature difference is between room and window. The thickness is the width of the air gap. Energy cost is found by multiplying the rate by time, converting to kWh, and applying the given utility rate. --- **Step 3: Step-by-Step Calculation** **1. Find window area:** \[ A = \text{height} \times \text{width} = 1.8\, \text{m} \times 1.0\, \text{m} = 1.8\, \text{m}^2 \] **2. Convert air gap thickness to meters:** \[ d = 5.75\, \text{mm} = 0.00575\, \text{m} \] **3. Temperature difference:** \[ \Delta T = 15^\circ \text{C} - 0^\circ \text{C} = 15\, \text{K} \] **4. Use formula for heat loss:** \[ Q = \frac{0.026\, \text{W/m·K} \times 1.8\, \text{m}^2 \times 15\, \text{K}}{0.00575\, \text{m}} \] \[ Q = \frac{0.026 \times 1.8 \times 15}{0.00575} \] \[ Q = \frac{0.702}{0.00575} \] \[ Q \approx 122\, \text{W} \] **5. Find total energy lost per week:** \[ \text{Energy (Wh)} = Q \times 24 \times 7 = 122\,\text{W} \times 168\, \text{h} = 20,496\, \text{Wh} = 20.5\, \text{kWh} \] **6. Calculate weekly cost:** \[ \text{Cost} = 20.5\, \text{kWh} \times 0.18\, \$/\text{kWh} = \$3.69 \] **Explanation block:** We first calculate the window's area, then convert the air gap thickness to meters. Next, we apply the convection formula using the temperature difference and the thermal conductivity of air. The result gives the power lost as heat. By multiplying this by the total hours in a week, we get the weekly energy lost. Finally, we convert this energy to kWh and multiply by the given utility rate for the final cost. --- **Final Answer:** The rate of heat loss through the window is **122 W**, and the associated weekly cost is **\$3.69**.