(a) What is the cost of heating a hot tub containing 1500 kg of water from 10.0^∘ C to 40.0^∘ C,

(a) What is the cost of heating a hot tub containing 1500 kg...

(a) What is the cost of heating a hot tub containing $1500 \mathrm{kg}$ of water from $10.0^{\circ} \mathrm{C}$ to $40.0^{\circ} \mathrm{C},$ assuming $75.0 \%$ efficiency to account for heat transfer to the surroundings? The cost of electricity is 9 cents/kW $\cdot$ h. (b) What current was used by the $220-\mathrm{V}$ AC electric heater, if this took $4.00 \mathrm{h}$ ?

Key Concepts

Specific Heat Capacity

This concept involves the amount of energy required to raise the temperature of a unit mass of a substance by one degree Celsius. It is crucial in calculating the energy needed to heat a given mass of water over a specific temperature change.

Thermal Efficiency

Thermal efficiency refers to the ratio of the useful energy transferred to the energy input, accounting for losses such as heat escaping to the surroundings. It is important in this context because only a portion of the total energy provided ends up heating the water.

Energy Unit Conversions

This involves converting energy from one unit, such as Joules, to another unit like kilowatt-hours, which is necessary for calculating costs based on energy prices. Understanding these conversions is key to bridging the gap between thermodynamic calculations and practical cost assessments.

Power, Energy, and Time Relationship

The relationship between power, energy, and time, typically expressed as energy equals power multiplied by time, is important for linking the energy consumption of an appliance with the operational time. This helps in determining overall energy costs and consumption over a specific period.

Electrical Current and Voltage

This concept describes how electrical power (the rate of energy consumption) relates to the current and voltage in an electrical circuit, typically through the equation P = VI. It is essential in determining the current drawn by an electrical heater when the power and voltage are known.

Transcript

00:01 Now the first part of this question involves finding what is the total cost of heating our hot tub.

00:11 And to find the total cost, we'll first need to find what is the total energy that is actually consumed to heat up our tub.

00:25 And we can find that by taking the heat capacity of water, multiplying by the mass of water, which is 1 ,500 kilograms, multiplying by the change in the temperature which is in this case 40 degrees minus 10 degrees so this gives us 30 degrees so this is the energy required to heat up our water the heat capacity of water is 4186 jous per kilogram per degree celsius and so this is the total amount of energy that is required now, to find the energy that is actually supplied, because we know that there's only 75 % efficiency.

01:34 So the total energy that is actually supplied must be 100%, right, taking this to be 75%, then the 100 % of the energy must be taking 1 .88 divided by 75 times a hundred.

01:56 So this is the energy that is being supplied by our heating, heating type, by the heating element.

02:15 We're given that the cost of electricity is 9 cents, 0 .09 per kilowatt hour.

02:31 What you want to find is the conversion from kilowatt hour to juice.

02:36 So 1 kilowatt hour.

02:39 Simply 1 ,000 watts times 3 ,600 seconds.

02:46 This is the number of seconds in an hour.

03:07 So dividing that we should be able to get what is the cost.

03:13 Taking 2 .51 times 10 power 8, divided by 3 .6 times 10 by 6.

03:18 This gives us the number of kilowatt hours that we have...