1. **Problem statement:** Calculate and compare the energy and cost of washing dishes using an electric dishwasher versus washing by hand, given water volumes, temperatures, and energy usage. 2. **Formulas and assumptions:** - Energy to heat water: $$Q = m c \Delta T$$ where $m$ is mass in pounds, $c = 1$ BTU/(lb\cdot°F) for water, and $\Delta T$ is temperature change. - 1 gallon of water = 8.0 pounds. - Dishwasher uses 0.500 kWh electrical energy. - 1 kWh = 3400 BTU. - Electricity cost = 0.11 per kWh. 3. **Calculate energy used by dishwasher:** - Water heated from 50°F to 140°F, so $\Delta T = 140 - 50 = 90$°F. - Mass of water: $10 \text{ gallons} \times 8 = 80$ pounds. - Energy to heat water: $$Q = 80 \times 1 \times 90 = 7200 \text{ BTU}$$ - Convert BTU to kWh: $$\frac{7200}{3400} = 2.1176 \text{ kWh}$$ - Total energy used by dishwasher: heating + electrical = $$2.1176 + 0.500 = 2.6176 \text{ kWh}$$ 4. **Calculate energy to heat water by hand:** - Water heated from 50°F to 100°F, so $\Delta T = 100 - 50 = 50$°F. - Mass of water: $20 \text{ gallons} \times 8 = 160$ pounds. - Energy to heat water: $$Q = 160 \times 1 \times 50 = 8000 \text{ BTU}$$ 5. **Calculate cost of electricity for each method:** - Dishwasher energy in kWh: 2.6176 kWh. - Dishwasher cost: $$2.6176 \times 0.11 = 0.288 \text{ (rounded)}$$ - Hand washing energy in kWh: $$\frac{8000}{3400} = 2.3529 \text{ kWh}$$ - Hand washing cost: $$2.3529 \times 0.11 = 0.259 \text{ (rounded)}$$ **Final answers:** - a. Dishwasher total energy used: $$2.62 \text{ kWh}$$ - b. Hand washing energy to heat water: $$8000 \text{ BTU}$$ - c. Cost of electricity: Dishwasher $$0.29$$, Hand washing $$0.26$$