max s. t. 6x₁ +8x2 + 5x3 + 9x4 2x1 + x₂ + x3 + 3x4 ≤ 5 x₁ + 3x₂ + x3 + 2x4 ≤ 3 1, 2, 3, 4 ≥0 Solve the LP using the simplex method.

A farmer has 500 feet of fencing and wants to build a pen that is rectangular in shape with three vertical subdivisions and one horizontal subdivision (see the diagram below). Note that his 500 feet of fencing must be used to make 8 lengths of fence (3 long, 5 short) as indicated by the black lines in the diagram. What are the outside dimensions of the pen that will give him the maximum area?

Farmer Ed has 750 meters of fencing, and wants to enclose a rectangular plot that borders on a river. If Farmer Ed does not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed? 750 -2x The width, labeled x in the figure, is meters. (Type an integer or decimal.) The length, labeled 750 – 2x in the figure, is meters. (Type an integer or decimal.) The largest area that can be enclosed is square meters (Type an integer or decimal.)

Find the dimensions of the rectangular box with largest volume such that the sum of the lengths of its twelve edges is 24. You may do this using either of the optimization techniques we learned in class, but you must justify your answer.