If the electric field inside a capacitor exceeds the dielectric strength of the dielectric between its plates, the dielectric will break down, discharging and ruining the capacitor. thus, the dielectric strength is the maximum magnitude that the electric field can have without breakdown occurring. the dielectric strength of air is 3.0 106 v/m, and that of neoprene rubber is 1.2 107 v/m. a certain air-gap, parallel plate capacitor can store no more than 0.090 j of electrical energy before breaking down. how much energy can this capacitor store without breaking down after the gap between its plates is filled with neoprene rubber?

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