A family comes home from a long vacation with laundry to do and showers to take. The water heater has been turned off during the vacation. If the heater has a capacity of 50.0 gallons and a 4 800-W heating element, how much time is required to raise the temperature of the water from 20.0 to 60.0? Assume the heater is well insulated and no water is withdrawn from the tank during that time.
Step 11 of 5
In this problem, water of volume is heated by a heating element of power . The water is heated from to . We calculate the time taken to heat the water.
Step 11 of 3
For the heating process we'll have to provide heat , which will be given by using power for time . Let's remember the definition of heating power:
The time required to create heat will therefore be
If heat is added to mass of a material of specific heat , the equation describing the situation will be
where and are the initial and final temperatures, respectively.
In our case, we don't know the mass of water, but we know its volume. Remembering the definition of density, we have
The mass will therefore be
Substituting, we have
Now we can substitute at the expression for the time, finding
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