Please answer all the questions from a to e. Thanks

(a) A machine shop makes pipes for the oil and gas industry in lengths of 250cm and 400cm. The shorter pipes give a profit of $140 each, while each long pipe gives a profit of $270. They can sell at most 900 of the 250cm pipes and at most 300 of the longer ones. Each pipe spends time on three machines as follows: For every long pipe made, there must be at least two short pipes made. For every pair of long pipes made, there can be at most seven short pipes made. Formulate an integer optimization model for the situation described above. (b) Solve the model from part (a) graphically (ignoring the need for integer values), clearly indicating the feasible region, and both the trial and optimal isovalue lines. Use algebra to determine the exact (non-integer) solution. (c) What is the best integer solution? (Verify feasibility!) (d) Suppose that things are the same as before but now they can purchase up to 500 minutes (all, none, or any fraction of this amount) of additional polishing time at a cost of $2 per minute, and can purchase an unlimited amount of extra grinding time at a cost of $3 per minute. Extend the model found in (a) by incorporating this new information. (e) Solve the model from part (d) using LINGO (use the scalar mode described in Chapter 2 of Decision Modeling)