In an air heating system, heated air at 40°c and 105 kpa absolute is distributed through a 0.2 m × 0.3 m rectangular duct made of commercial steel at a rate of 0.5 m3/s. determine the pressure drop and head loss through a 30-m-long section of the duct. the roughness of commercial steel surfaces is ε = 0.45 m. the dynamic viscosity of air at 40°c is μ = 1.918 × 10–5 kg/m·s, and it is independent of pressure. the pressure drop is pa. the head loss is m.

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