olve the standard minimization problem using duality. (You may already have seen some of them in earlier sections, but now you will be solving them using a different method.) Minimize c = s + t subject to s + 8t ≥ 54 8s + t ≥ 54 s ≥ 0, t ≥ 0. c= (s, t) =

Maximize p = 2x + y subject to x + 2y ≤ 12 −x + y ≤ 8 x + y ≤ 8 x ≥ 0, y ≥ 0.

For a certain company, the cost function for producing a items is C (x) = 30 z + 150 and the revenue function for selling a items is R(x) = -0.5(x – 70)2 + 2,450. The maximum capacity of the company is 100 items. The profit function P(æ) is the revenue function R(r) (how much it takes in) minus the cost function C (r) (how much it spends). In economic models, one typically assumes that a company wants to maximize its profit, or at least make a profit! Answers to some of the questions are given below so that you can check your work. 1. Assuming that the company sells all that it produces, what is the profit function? P(z) =| Hint: Profit = Revenue - Cost as we examined in Discussion 3. 2. What is the domain of P (x)? Hint: Does calculating P(x) make sense when a = -10 or x = 1,000? 3. The company can choose to produce either 40 or 50 items. What is their profit for each case, and which level of production should they choose? Profit when producing 40 items = Number Profit when producing…

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