If a water heater has a capacity of 54.4 gallons and a 4910 W heating element, how much time is required to raise the temperature of the water from 19.8oC to 60.0oC? Assume that the heater is well insulated and no water is withdrawn from the tank during this time.

1 gallon of water has a mass of 3.785 kg. Therefore you are heating

M = 54.4*3.785 = 205.9 kg. The total enegy required to heat that amount of water 40.2 C is
205.9 kg*40.2 C*4184 Cal/kg*C
= 3.463*10^7 J

Divide that by the watts of the heater (4910) to get the required time in seconds.

To calculate the time required to raise the temperature of the water from 19.8°C to 60.0°C, we can use the formula:

Heat gained by water = Power × Time × Efficiency

In this case, the heat gained by the water should equal the heat provided by the heating element.

First, let's calculate the heat gained by the water:

Heat gained by water = Mass × Specific heat capacity × Temperature change

To calculate the mass of water, we can use the formula:

Mass = Volume × Density

Given that the water heater has a capacity of 54.4 gallons, we need to convert it to liters (1 gallon = 3.78541 liters):

Volume = 54.4 gallons × 3.78541 = 205.58744 liters

The density of water is approximately 1 gram per milliliter or 1000 kilograms per cubic meter.

Density = 1000 kg/m^3

Now we can calculate the mass of the water:

Mass = 205.58744 liters × 1000 kg/m^3 = 205,587.44 grams = 205.58744 kg

The specific heat capacity of water is approximately 4.186 joules per gram degree Celsius.

Specific heat capacity = 4.186 J/g°C

Now we can calculate the heat gained by the water:

Heat gained by water = 205.58744 kg × 4.186 J/g°C × (60.0 - 19.8)°C = 335,950.6536 J

Next, we can substitute this value into the equation:

335,950.6536 J = 4910 W × Time × Efficiency

Assuming the efficiency of the heater is 100% (all the power is transferred to the water), then the equation becomes:

335,950.6536 J = 4910 W × Time

To calculate the time (in seconds), we rearrange the equation:

Time = 335,950.6536 J / 4910 W

Time = 68.495907 seconds

Therefore, it would take approximately 68.5 seconds to raise the temperature of the water from 19.8°C to 60.0°C.