Question

The label on a washing machine indicates that the washer will use \(\$ 85\) worth of hot water if the water is heated by a 90 percent efficient electric heater at an electricity rate of \(\$ 0.09 / \mathrm{kWh}\). If the water is heated from 18 to \(45^{\circ} \mathrm{C}\), the amount of hot water an average family uses per year is \((a) 11.6\) tons (b) 15.8 tons \((c) 27.1\) tons (d) 30.1 tons \((e) 33.5\) tons

Short answer

(Note: 1 ton = 1000 kg) Step 1: Calculate the energy expenditure on water heating. Energy_consumed (kWh) = $85 / (0.9 × $0.09 / kWh) Step 2: Calculate the energy required to heat 1 kg of water from 18°C to 45°C. Energy (kWh) = (1 × 4186 × 27) / 3600000 Step 3: Calculate the total mass of water that can be heated in a year. Total_mass (kg) = Energy_consumed (kWh) / Energy_per_kg (kWh) Step 4: Convert the mass of water from kg to tons and choose the correct option. Total_mass (tons) = Total_mass (kg) / 1000 Using the given information, calculate the energy expenditure, the energy required to heat 1 kg of water, and the total mass of water that can be heated in a year. Then, convert the mass of water from kg to tons and choose the correct option.

Step-by-step solution

Calculate the energy expenditure on water heating.

The electric heater is 90% efficient, and the cost of electricity is \(\$0.09 / \mathrm{kWh}\). We want to find the energy consumed in kWh using the given cost. So, we have: Energy_consumed (kWh) = \(\frac{Cost}{Efficiency × Electricity\_rate}\) Energy_consumed (kWh) = \(\frac{\$85}{0.9 × \$0.09 / \mathrm{kWh}}\)

Calculate the energy required to heat water from 18°C to 45°C.

Now let's calculate the amount of energy required to heat 1 kg of water from 18°C to 45°C. We can use the formula: Energy (kWh) = \(\frac{Mass × Specific\_heat × Temperature\_change}{3600000}\) where mass is in kg, specific_heat is in \(\frac{J}{kg \times K}\) and temperature_change is in K. The specific heat of water is approximately 4,186 J/kg×K. Temperature_change = 45°C - 18°C = 27 K For 1 kg of water: Energy (kWh) = \(\frac{1 × 4186 × 27}{3600000}\)

Calculate the total mass of water that can be heated in a year.

Now, let's find the total mass of water that can be heated in a year with the calculated energy consumption. For this, we will divide the annual energy_consumed by the energy required to heat 1 kg of water: Total_mass (kg) = \(\frac{Energy\_consumed (kWh)}{Energy\_per\_kg (kWh)}\)

Convert the mass of water from kg to tons and choose the correct option.

Finally, convert the mass of water from kg to tons: Total_mass (tons) = \(\frac{Total\_mass (kg)}{1000}\) Compare this value to the given options and choose the correct answer.