Chapter 3.6, Problem 52E

Power and energy Power and energy are often used interchangeably, but they are quite different. Energy is what makes matter move or heat up. It is measured in units of joules or Calories, where 1 Cal = 4184 J. One hour of walking consumes roughly 10⁶ J, or 240 Cal. On the other hand, power is the rate at which energy is used, which is measured in watts, where 1 W = 1 J/s. Other useful units of power are kilowatts (l kW = 10³ W) and megawatts (l MW = 10⁶ W). If energy is used at a rate of l kW for one hour, the total amount of energy used is 1 kilowatt-hour (l kWh = 3.6 × 10⁶ J). Suppose the cumulative energy used in a large building over a 24-hr period is given by $E(t)=100t+4t^2-\frac{t^3}{9}\text{kWh}$, where t = 0 corresponds to midnight.

1. a. Graph the energy function.
2. b. The power is the rate of energy consumption; that is, P(t) = E′(t). Find the power over the interval 0 ≤ t ≤ 24.
3. c. Graph the power function and interpret the graph. What are the units of power in this case?