[1]

(a)(part of #23 Pg 241) (3 pts) Using arrows attached to each of the 5 lines you graph, show the solution set for each of the 5 inequalities of the system in the box below ; all graphs must be drawn on the same set of axes (as at the right), without using a graphing calculator. REVIEW THESE IDEAS? (b)(2 pts) On the same graph (as at the right) and again for the system named in the box below, CLEARLY shade in the solution set for the entire system, using the side of a pencil or similar instrument: do not hide your answer to [1a].

[2]

(a)(#23 Pg 241, cont'd) (3 pts) Write the coordinates of all 5 corner points of the solution set on your graph in [1b] above. (b)(1 pt) State the value of the objective function z = 20x + 30y at each of the 5 corners in [2a] above, using the table below, (c)(1 pt) Name the corner point at which z has its biggest value.

2x + 10y 80 6x + 2y 72 3x + 2y 6   x 0   y 0

[3]

(Pg 270 #15) Translate the problem in the box below into a system of constraints (i.e., inequalities) and an objective function. Define all variables clearly in English : recall that "Let x = regular cola" was not a clear definition on test 3. DO NOT SOLVE the translated problem; just "set it up".

Jon includes 3 foods on Garfield's menu: peas, bread, and steak. These foods contain nutrients (oops!) as follows :     We do this 6 times a day, right Jon?? calories protein fat 1 ounce of peas 20 0.5 grams 1.5 grams 1 ounce of bread 50 1.0 grams 3.0 grams 1 ounce of steak 56 9.0 grams 2.0 grams Garfield must consume no more than 1000 calories and no more than 35 grams of fat. Determine the number of ounces of each food that Garfield should eat to maximize the amount of protein eaten.

[4]

(parts of #5 Pg 269, #17 and #21 Pg 325) For the linear programming problem in the box (at the right), do as follows : For each of the 6 lines of the problem in the box, write either NON-STANDARD or STANDARD, depending upon whether or not one of the four conditions for a standard problem are violated. These 4 conditions appear above problem [7] on this test.

Minimize w = 15x + 8y subject to : -3x + y 8 3x + 2y 48 4x + 6y 84 x 0 y 0 line 1 ___________   line 2 ___________ line 3 ___________ line 4 ___________   line 5 ___________ line 6 ___________

[5]	(#21 Pg 289, one number changed to make problem [6] easier) Write the SIMPLEX TABLEAU ( = ST ) for the problem at the right but then STOP : DO NOT SOLVE.	Maximize z = 2x1 + x2 subject to:  3x1 +  2x2 22  3x1 +  4x2 34   x1 0   x2 0
[6]	(rest of simplified #21 Pg 289) Use THE SPECIFIC STEPS OF SIMPLEX METHOD OUTLINED IN OUR CLASS HANDOUTS to solve problem [5] (in box at right). WITH CORRECT WORK, you will encounter one pivot and a few fractions (all with denominator 3) : the original problem needed 2 pivots. See [C], [E], & [F] at top of test ; clearly write all row operations BETWEEN the affected matrices, using the name format described below.

Row operation names must be either rn + k rm = Rn or      k rn = Rn , where Rn is the name of the (new) row being built, and rn or rm are the name(s) of rows in the (old) existing matrix. For example, if m=1, n=2, and k=4, we write : r2 + 4 r1 = R2 . Each step of your work in problems [6] and [8] should appear as shown below, as it did earlier in Gauss/Jordan.

old matrix

row op name

new matrix

[7] [8] [9]

Minimize w = 15x1 + 8x2 subject to: x1 + 2x2 20 3x1 + 2x2 36 x1 0 x2 0

for use with [7]-[9] below These criteria must be met by standard maximizing problems C1. The objective function is to be maximized C2. All non-objective variables must be 0 C3. All inequalities must be of the type (as in [5] above) C4. All right hand constants must be 0 Problem in box at left is #17 Pg 325

[7]	[a](1 pt) Convert problem [7] (above box) into a problem satisfying C1, C2, and C3 (see above). [b](2 pts) Write your answer to [7a] above as a set of equations with slack variables s1 and s2 [c](2 pts) Write the simplex tableau of your converted problem in [7b] above.

[8]

(6 points) Use THE SPECIFIC STEPS OF SIMPLEX METHOD OUTLINED IN OUR CLASS HANDOUTS, to solve the problem in [7] above. Name row operations as required in [6] above. With correct work, you will encounter 2 pivots, one in the first phase of the simplex method, and one in the second phase, at which point you will arrive at a FINAL TABLEAU. Take care in locating indicator and pivot rows. Again note [C],[E], and [F] at the top of this test. All fractions in your matrices will have denominators 2, 3, and 4 (no others) ; one arithmetic calculation will use denominator 6 .

[9]

(4 pts) You will encounter 3 simplex tableaus in [8] above, not counting intermediate matrices. The basic solution for the 1st tableau (in [7c]) is is:   x1 = x2 = 0, s1 = 20, s2 = -36, z = w = 0. Name the values of each of these variables (here or in blue book) for the 2nd and 3rd tableaus : see below.

1st tableau from [7c] basic solution x1=x2=0 s1 = 20, s2 = -36, z=w=0

1st pivoting

2nd tableau x1 = x2 = s1 = s2 = z   = w  =

2nd pivoting

3rd tableau x1 = x2 = s1 = s2 = z   = w  =