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 represented by the estimate inverse demand function P = 1,200 - 4Q,

marginal revenue function is MR = 1,200 - 8Q,

the utility company has set up two electricity 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 estimated as

C₁(Q₁) = 8,000 + 6Q₁² and MC₁ = 12Q₁ and

C₂(Q₂) = 6,000 + 3Q₂² and MC₂ = 6Q₂, respectively.

Determine the profit maximizing amounts of electricity to produce at the two facilities, the optimal price, and the utility company's profits. Please show your calculations.