Power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up and is measured in units of joules (J) or Calories (Cal), where 1 Cal = 4184 J. One hour of walking consumes roughly 10⁶ J, or 250 Cal. On the other hand, power is the rate at which energy is used and is measured in watts (W; 1 W = 1 J/s). Other useful units of power are kilowatts (1 kW = 10³ W) and megawatts (1 MW = 10⁶ W). If energy is used at a rate of 1 kW for 1 hr, the total amount of energy used is 1 kilowatt-hour (kWh), which is 3.6 × 10⁶ J. 

Suppose the power function of a large city over a 24-hr period is given by

where P is measured in megawatts and t = 0 corresponds to 6:00 p.m. (see figure).

a. How much energy is consumed by this city in a typical 24-hr period? Express the answer in megawatt-hours and in joules.

b. Burning 1 kg of coal produces about 450 kWh of energy. How many kg of coal are required to meet the energy needs of the city for 1 day? For 1 year?

c. Fission of 1 g of uranium-235 (U-235) produces about 16,000 kWh of energy. How many grams of uranium are needed to meet the energy needs of the city for 1 day? For 1 year?

d. A typical wind turbine can generate electrical power at a rate of about 200 kW. Approximately how many wind turbines are needed to meet the average energy needs of the city?

Transcribed Image Text:

P(t) = E'(t) = 300 – 200 sin 12 P(f) = E'(f) = 300 – 200 sin 600 12 400 200 12 24 Time (hr) Power (MW)