A refrigerator cycles on and off. Let's say it consumes electrical power at a rate of \(150 \mathrm{~W}\) when it's on, and (essentially) \(0 \mathrm{~W}\) when it's off. If it spends half of its time in the on-state, what is its average power? How much energy does it consume in a 24 -hour day, in kWh? At a typical electricity cost of \(\$ 0.15\) per \(\mathrm{kWh}\), about how much does it cost per year to run the refrigerator?

Energy consumption calculation is a fundamental concept for anyone looking to understand their electrical appliance usage and its impact on the electricity bill. Consider a refrigerator, an appliance that turns on and off throughout the day. It might consume 150 W of power when on, and nearly 0 W when off. If it spends an equal amount of time in both states within a period, say 12 hours on and 12 hours off in a 24-hour day, how do we calculate the average energy consumption?

To determine average energy usage, we need to look at the total energy consumed over a specific time period and divide that by the time. Energy is power consumed over time, typically measured in watt-hours (Wh) or kilowatt-hours (kWh), with 1 kWh equaling 1,000 Wh. Since our refrigerator uses 150 W for half of the day, its average power use over a 24-hour period would be half of 150 W, which is 75 W. In terms of energy consumption for the entire day in kWh, one would multiply the average power (converted to kW by dividing by 1,000) by the number of hours in a day, resulting in the daily energy consumption. Thus, the answer to our exercise would be:

The electrical power rate conveys how fast an appliance like a refrigerator uses energy and is expressed in watts (W). A power rate of 150 W means that when the refrigerator is actively cooling, it consumes energy at a rate that would amount to 150 joules every second. Understanding the power rate is a cornerstone for effectively managing energy usage and, consequently, costs. For instance, knowing the power rate, we can assess the efficiency of appliances and decide on actions to save on energy bills.

When discussing the time an appliance spends in active mode, this directly affects the average power consumption. Over a day, if the refrigerator's active time is 50% of the total time, it doesn't mean it uses 150 W all day. Instead, the average rate of energy consumption is reduced accordingly, to 75 W as shown in the exercise. The power rate, when multiplied by time, provides us with the total energy used, which can be analyzed for cost-effectiveness, a segue into our next concept, cost analysis.

Cost analysis in the context of energy consumption involves calculating the total cost incurred from using an appliance over a certain period. To assess this for our refrigerator example, you first need to determine the total energy consumption over that period, which is found by multiplying the average power by the time the refrigerator is in use. With the energy usage figured out, the cost can be ascertained by applying the appropriate electricity rate. In many areas, electricity is billed in cents per kWh used.

To extrapolate the yearly cost, you multiply the daily energy usage in kWh by the cost rate and then by the number of days in a year. The result is a tangible figure, in dollars, that represents the annual expense of using the refrigerator. Hence, the step-by-step solution concludes that the yearly cost to run the refrigerator in our exercise is approximately $98.55, derived from the daily energy consumption mulitplied by the cost rate and days in the year.