1. Make a sketch of the feasibility region defined by the following constraints. Label the edges of the region with numbers; label the extrema with letters. Find and present the coordinates of the extrema. Assume that x and y are both equal to or greater than zero.
3y<=4x, 2x+3y<=8, x>=3
2. The constraints on a particular manufacturing process are shown on the right. The extrema of the feasibility region have been calculated and plotted.
Using the profit function given below, calculate the profit (value of P) at each extrema.
At which extremum is the profit the maximum? The minimum? (A negative profit is a loss. The minimum profit is either the smallest positive profit, or the largest loss.)
3. Eye-Full Optics assembles astronomical telescopes (x), premium binoculars (y) and student-grade microscopes (z) from imported parts. Each telescope takes one hour to assemble, each pair of binoculars two hours, and each microscope four hours; the availability of skilled labor limits assembly work to 1000 hours per day. Eye-Full has a contract with FedEx, and must ship no less than 400 items per day. A contract with a major retailer requires them to deliver a minimum of 100 telescopes, 250 binocs, and 50 microscopes per day. But there are supply limitations. The telescopes and binocs are shipped with the same eyepieces; each scope has one, and each pair of binocs has two. The subcontractor who supplies the eyepieces can only furnish 800 per day. Similarly, both the binocs and the microscopes use the same prisms; each pair of binocs needs two, and each microscope needs four. The prism supplier can only ship Eye-Full 1600 per day.
If Eye-Full makes a profit on $100 on each scope, $200 on each pair of binocs, and $350 on each microscope, how many of each should the company manufacture each day? What is its daily profit?
(Since the feasibility region is a volume in three-dimensional space, a sketch is not required.)
• Graphics must be neat, clear and complete. A graphics app can be used, but a freehand sketch is also acceptable.
• All calculations should be shown.
• All answers must be clearly stated.
• Relevant theory should be cited as necessary to explain which procedures were used to arrive at the answers, and why.
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