MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
During a 24-hour period, the power output of a wind farm is modeled by a continuous, non-negative function p, where p(t) is measured in megawatts and t in hours (0≤t≤24). By the end of the day, the total energy generated is known to be 480 megawatt-hours.
(a) Determine the average power output, in megawatts, produced by the wind farm during the 24-hour period. Include units in your answer.
(b) Company planners wish to know how long it takes, starting at midnight (t=0), for the average power output to reach 24 megawatts. Let E(t)=∫0tp(u)du denote the total energy produced by time t. Write an equation involving E whose solution gives the desired time T, and explain why that equation is valid.
(c) Suppose additional monitoring shows E(t)=5t3/2 for 0≤t≤24. Solve for T in hours using your equation from part (b), and interpret the result in the context of the wind farm’s operation.