Power and energy are often used interchangeably, but they are quite different. Energy is measured in units of joules ( J ) or Calories (Cal), where \( 1 \mathrm{Cal}=4184 \mathrm{~J} \). On the other hand, power is the rate at which energy is used and is measured in watts (W; \( 1 \mathrm{~W}=1 \mathrm{~J}) \). Other useful units of power are kilowatts \( (\mathrm{kW}) \) and megawatts \( \left(1 \mathrm{MW}=10^{6} \mathrm{~W}\right. \) ). If energy is used at a rate of 1 kW for 1 hr , the total amount of energy used is 1 kilowatt-hour \( (\mathrm{kWh}) \), which is \( 3.6 \times 10^{6} \mathrm{~J} \). Suppose the power function of a large city over a 24 -hr period is given by \( \mathrm{P}(\mathrm{t})=\mathrm{E}^{\prime}(\mathrm{t})=345-230 \sin (\mathrm{xt} / 12) \), where P is measured in MW and \( \mathrm{t}=0 \) corresponds to \( 6: 00 \mathrm{p} . \mathrm{m} \). (see accompanying figure). Complete parts (a) through (d) below. Click the icon to view the graph of \( \mathrm{P}(\mathrm{t})=\mathrm{E}^{\prime}(\mathrm{t})=345-230 \sin (\mathrm{xt} / 12) \). The amount of uranium needed in a day is 518 g . (Simplify your answer. Round to the nearest integer as needed.) The amount of uranium needed in a year is 189070 g . (Simplify your answer.) d. A typical wind turbine can generate electricity at a rate of about 200 kW . Approximately how many wind turbines are needed to meet the average energy needs of the city? The number of wind turbines needed to meet the average energy needs of the city is (Simplify your answer.).
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The Deep Dive
Energy has played a vital role in human civilization since the dawn of time. The use of resources like coal, oil, and later renewable energies has radically transformed our societies from the agricultural age to the industrial and now the digital age. The difference between power and energy is crucial in understanding these transformations. As societies evolved, so did their energy consumption patterns—leading to innovations in energy efficiency and the advent of sustainable energy sources, which are more important than ever in today’s climate-conscious world! In real-world applications, knowing the relationship between power, energy, and time helps us make smarter choices about energy consumption and production. For instance, if you’re planning a solar installation, understanding how many kilowatt-hours your home uses can guide your choice on the number of panels you’ll need. Similarly, knowing that a wind turbine produces around 200 kW of power can help cities forecast and meet their energy demands, pushing them toward cleaner and more sustainable energy solutions, making a brighter future for all!
