Question 1. Consider the following LP problem: min ACR s.t 2= 2x₁ +5x₂ 2 5, 9, -4, 0. A VIII -2x₁ + x₂ = x1, x₂ 2 min z = 2x₁ + 5x₂ s.t x1 <-X3 S Convert it into standard form. Check ift = (4.) is a feasible solution. If it is, then construct one feasible solution of the standard form from this solution. Solution: We only need to add a surplus variable, for the first constraint, and a slack variable, for the second constraint. -2x₁ + x₂ X.X.X.20 + x₁ = We can check thatx = (4.) is not a feasible solution of this problem since it does not satisfy the constraint -2x1 + x2 = -4.

Suppose the coal and steel industries form a closed economy. Every $1 produced by the coal industry requires $0.30 of coal and $0.70 of steel. Every $1 produced by steel requires $0.80 of coal and $0.20 of steel. Find the annual production (output) of coal and steel if the total annual production is $20 million.

If during the following year it is predicted that each comedy skit will generate 30 thousand and each musical number 20 thousand, find the maximum income for the year. A television program director must schedule comedy skits and musical numbers for prime-time variety shows. Each comedy skit requires 2 hours of rehearsal time, costs 3000, and brings in 20,000 from the shows sponsors. Each musical number requires 1 hour of rehearsal time, costs 6000, and generates 12,000. If 250 hours are available for rehearsal and 600,000 is budgeted for comedy and music, how many segments of each type should be produced to maximize income? Find the maximum income.

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