Question: 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 \(P=1,000-5Q\), the utility company has set up two electric generating facilities: \(Q_{1}\) kilowatts are produced at facility 1, and \(Q_{2}\) kilowatts are produced at facility 2 (so \(\left.Q=Q_{1}+Q_{2}\right)\). The costs of producing electricity at each facility are given by \(C_{1}\left(Q_{1}\right)=10,050+5 Q_{1}^{2}\) and \(C_{2}\left(Q_{2}\right)=5,000+2 Q_{2}^{2}\), respectively.

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

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