Question

An air conditioner whose coefficient of performance is 2.00 removes $1.73 \times 10^{8} \mathrm{J}$ of heat from a room per day. How much does it cost to run the air conditioning unit per day if electricity costs \(\$ 0.10\) per kilowatt-hour? (Note that 1 kilowatt-hour \(=3.6 \times 10^{6} \mathrm{J} .\) )

Short answer

Answer: To find the daily cost of running the air conditioning unit, calculate the final value using the formula: Daily cost = \(( (1.73 \times 10^{8} \mathrm{J}) / 2.00) \cdot (1 \mathrm{kWh}) / (3.6 \times 10^{6} \mathrm{J}) \times \$0.10/\mathrm{kWh}\) .

Step-by-step solution

Calculate work done by the air conditioner per day (W)

We will use the formula for the coefficient of performance (COP) of the air conditioner to find the work done: COP = (Heat removed per day) / (Work done per day) Work done per day (W) = (Heat removed per day) / COP W = \((1.73 \times 10^{8} \mathrm{J}) / 2.00\)

Convert work done per day (W) to kilowatt-hours (kWh)

Now, convert the work done per day (W) from Joules to kilowatt-hours using the given conversion: 1 kilowatt-hour = \(3.6 \times 10^{6} \mathrm{J}\) W (kWh) = \((1.73 \times 10^{8} \mathrm{J}) / 2.00 \cdot (1 \mathrm{kWh}) / (3.6 \times 10^{6} \mathrm{J})\)

Calculate the daily cost of running the air conditioner

Finally, calculate the daily cost by multiplying the kilowatt-hours (kWh) with the electricity cost per kilowatt-hour: Daily cost = W (kWh) × cost per kilowatt-hour Daily cost = \(( (1.73 \times 10^{8} \mathrm{J}) / 2.00) \cdot (1 \mathrm{kWh}) / (3.6 \times 10^{6} \mathrm{J}) \times \$0.10/\mathrm{kWh}\) Calculate the final value to get the cost of running the air conditioning unit per day.