A uniform 4.00 kg square solid wooden gate 2.00 m on each side hangs vertically from a frictionless pivot at the center of its upper edge. A 1.30 kg raven flying horizontally at 5.00 m\s flies into this door at its center and bounces back at 2.50 m/s in the opposite direction. What is the angular speed of the gate just after it is struck by the unfortunate raven?

The second largest public utility in the nation is the sole provider of electricity in 32 counties of southern florida. to meet the monthly demand for electricity in these counties, which is given by the inverse demand function p = 1,200 – 4q, the utility company has set up two electric generating facilities: q1 kilowatts are produced at facility 1, and q2 kilowatts are produced at facility 2 (so q = q1 + q2). the costs of producing electricity at each facility are given by c1(q1) = 8,000 + 6q12 and c2(q2) = 6,000 + 3q22, respectively. determine the profit-maximizing amounts of electricity to produce at the two facilities, the optimal price, and the utility company’s profit