26. The energy economy of an area is composed of four industries: gas, coal, hydroelectric, and nuclear. The three main consumers of energy are area residential consumers, a manufacturing plant, and a university. Assume that each consumer may use some of the energy produced by each industry, and also that each industry uses some of the energy produced by each of the other industries. The energy needs of each consumer and industry is represented by a demand vector whose entries, in order, give the amount of gas, coal, hydroelectric, and nuclear energy needed by each consumer or industry, in some convenient units. The demand vectors for the consumers are: DR = 2401; DM = 3025; DU = 0351; and the demand vectors for the industries are: DG = 0132; DC = 4012; DH = 3101; DN = 2130. The price of gas is $15,000 per unit, the price of coal is $10,000 per unit, the price of hydroelectric power is $7000 per unit, and the price of nuclear energy is $9000 per unit. These prices can be represented by the (column) price vector:
Find the income earned by the gas industry and its cost for the other forms of energy it uses. Then calculate its profit.