Let's break down the problem step by step as requested. --- ## **Step 1: Heat Required to Bring Ice from -20°C to 40°C** need to calculate the total heat (\ Q \)) required to: 1. Warm ice from -20°C to °C. 2. Melt the ice at °C to water. 3. Warm the resulting water from °C to 40°C. Let's use these constants: - Specific heat of ice, \( c_{ice} = 2.1 \text{ kJ/kg} \cdot \text{°C} = 210 \text{ J/kg} \cdot \text{°C} \) - Latent heat of fusion of ice, \( L_f = 334 \text{ kJ/kg} = 334,000 \text{ J/kg} \) - Specific heat of water, \( c_{water} = 4.18 \text{ kJ/kg} \cdot \text{°C} = 418 \text{ J/kg} \cdot \text{°C} \) - Mass of ice, \( m = .200 \) kg ### **Step 1A: Warm Ice from -20°C to °C** \[ Q_1 = m \cdot c_{ice} \cdot \Delta T = .2 \cdot 210 \cdot (-(-20)) = .2 \cdot 210 \cdot 20 = 8,400 \text{ J} \] **Explanation:** This is the heat needed to raise the temperature of the ice from -20°C to °C. --- ### **Step 1B: Melt the Ice at °C** \[ Q_2 = m \cdot L_f = .2 \cdot 334,000 = 66,800 \text{ J} \] **Explanation:** This is the heat needed to melt the ice at °C to water. --- ### **Step 1C: Warm Water from °C to 40°C** \[ Q_3 = m \cdot c_{water} \cdot \Delta T = .2 \cdot 418 \cdot (40-) = .2 \cdot 418 \cdot 40 = 33,440 \text{ J} \] **Explanation:** This is the heat needed to raise the temperature of the water from °C to 40°C. --- ### **Step 1D: Total Heat Required** \[ Q_{total} = Q_1 + Q_2 + Q_3 = 8,400 + 66,800 + 33,440 = 108,640 \text{ J} \] --- ## **Step 2: Time Required to Bring Water Up to 40°C** Given: - Power of heater, \( P = 150 \) W \( = 150 \) J/s \[ t = \frac{Q_{total}}{P} = \frac{108,640}{150} \approx 724.3 \text{ s} \] **Explanation:** This is the time required for the 150 W heater to supply the calculated total heat to the system. --- ## **Final Answers** 1. **Heat required to bring the ice up to 40°C:** \( \boxed{108,640 \text{ J}} \) 2. **Time required:** \( \boxed{724 \text{ s}} \) (rounded to the nearest second) --- If you want to proceed to the next steps (final temperature after adding aluminum rod, change in length), let me know!