Final answer:
To heat twenty gallons of water from 70°F to 120°F, you need to input 8,340 Btu of energy. Depending on the efficiency of the heater (78%, 85%, or 90%), the energy you need to supply is 10,692.31 Btu, 9,811.76 Btu, and 9,266.67 Btu, respectively.
Step-by-step explanation:
To calculate the amount of natural gas needed to heat twenty gallons of water from 70°F to 120°F, we need to know the energy required for heating the water and the efficiency of the water heater.
The amount of heat (Q) required to raise the temperature of water can be calculated with the formula Q = m⋅C⋅ΔT, where m is the mass of the water, C is the specific heat capacity of water, and ΔT is the change in temperature.
First, convert gallons to pounds since energy formulae typically use mass (1 gallon of water is approximately 8.34 pounds), then calculate the energy needed:
Q = 20 gallons ⋅ 8.34 pounds/gallon ⋅ 1 Btu/(pound°F) ⋅ (120°F - 70°F)
Q = 166.8 pounds ⋅ 50°F ⋅ 1 Btu/(pound°F) = 8,340 Btu
Next, calculate the energy that must be input into the water heater to account for its inefficiency:
E = Q / Efficiency
- For an efficiency of 78%: E = 8,340 Btu / 0.78 = 10,692.31 Btu
- For an efficiency of 85%: E = 8,340 Btu / 0.85 = 9,811.76 Btu
- For an efficiency of 90%: E = 8,340 Btu / 0.90 = 9,266.67 Btu
These values represent the energy that needs to be supplied by the natural gas for each efficiency level to reach the desired temperature.