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Math 76O-143 Finite Mathematics

(Inequalities and Simplex Method)
Submit on the internet by 11:59 PM on Sunday 13 March 2016
Maximum value toward semester grade is 4 pts
Methods discussed in class using other problems.
Links to other web pages were not on original test;
CAUTION: Prof McFarland makes new tests each semester.

Click or type the appropriate answer for each question.
 Most of myfriends didwell onthis stuff.
 Average grade on Fall 2015 paper test #4 : 32.92
[1]
At the right are 5 lines: the 2 axes and 3 others in color. These lines include the graphs of the equations associated with the inequalities boxed below. Click any or all boxes(s) which are located in the solution set of the system of inequalities boxed below.
On paper tests, students will be asked to create these graphs without a graphing calculator.
 x + 2y 62 4x + y 66 x 0 ; y 10
[2]
 (a)(3 points) Find the coordinates of all 3 corner points of the solution set which you identified as an answer to [1] above. Enter these coordinates in the top row below. (b)(2 points) Find the value of the objective function [P = x + 3y] at each of the 3 corners in [2a] above. Enter these values in the bottom row below. Scoring : You must first earn a perfect 5 pts in [1] and 3 pts in [2a], 1 pt for each corner point. [2b] is then scored "all or nothing" (c) (unscored) Note where P has its biggest value.
 Corner point with smallest x-coordinate Corner point with biggest x-coordinate Corner Point Coordinates : ( x , y ) (  ,  ) (  ,  ) (  ,  ) Value of P = x + 3y

[3] Consider the problem in the box below : without solving it.
 The Kane Company makes two models of grills, one gas and one electric. Each gas grill requires 3 pounds of iron and 6 minutes of labor; each electric grill requires 4 pounds of iron and 3 minutes of labor. Kane must produce at least 150 gas grills to fill a backlog of orders for this model only. Each gas grill yields a profit of \$2, and each electric grill yields a profit of \$3. Kane has a supply of 1000 pounds of iron, and 20 hours of labor available. How many grills of each type should Kane produce in order to maximize it's profit?
 Just set this up.
 SEE A SIMILAR EXAMPLE ?
(a)(3 pts) In the box below are various ways to define variables in solving the above problem. Choose one or more of these definitions of variables, so that taken together, your choices would be the best in solving the problem above.
On paper tests, you must write definitions rather than choose among options.
Scoring: 2 points for variables x and y, one point for the objective function (capital letter)
 Let x = number of pounds of iron used Let y = number of minutes of labor used Let x = profit on the gas grills Let y = profit on the electric grills Let x = the 1000 pounds of iron available Let y = the 20 hours of labor available Let x = gas grills Let y = electric grills Let x = number of gas grills produced Let y = number of electric grills produced Let x = \$2 profit on each gas grill Let y = \$3 profit on each electric grill Let A = total amount of iron used in both types of grill Let M = total number of minutes needed Let P = total profit on sales of both types of grill Let P = profit
(b)(2 pts) Using the best definition of variables from the choices in [3a] above, choose one or more of the options below to assemble a correct translation of the original problem into a set of inequalities and objective function below.
You must first earn a perfect 3-point score on [3a] above in order to earn points in [3b].
Scoring below : Inequalities worth 1½ points, objective function worth ½ point

 x 1000 y 20 x = 2 y = 3 x 150 x 150 y 1200 6x + 3y 1200 3x + 4y 1000 x + y = 1000 x 0 y 0 x 0 y 0 A = 3x + 4y M = 6x + 3y P = 2x + 3y P = x + y

 [4] 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, click 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 C = 8x + 12ysubject to: Standard Non-standard x + 3y 2 Standard Non-standard 2x + 2y 3 Standard Non-standard x + 2y 12 Standard Non-standard x 0 Standard Non-standard y 0 Standard Non-standard

I can help

[5] Enter numbers of the SIMPLEX TABLEAU ( = ST ) for the problem at the right. Continued in [6] below.
Scoring : bottom row worth 3 points, remaining 2 rows worth 1 point each
Maximize P = 4x1 + 6x2
subject to :
 x1 + 2x2 24 3x1 + x2 8 x1 0 x2 0
ST =
[6]
 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 no fractions. As in our text, two Slack Variables will be named s1 and s2 . On tests given in class, the test-taker must also write the matrices, since without them, the 2 needed row operations and basic solution ( 7 answer boxes below) cannot be discovered. Scoring : You must get a perfect (5 pt) score on [5] above before trying [6]. Then for [6] : 1 point for each row operation (left box) ; 1 point for each row of the basic solution (right box)
 startingtableau
r1 + r2 = R1

r3 +
r2 = R3
 Finaltableau
Final basic solution :

 x1 =  ;   x2 = s1 =   ;   s2 = P =

 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. Each step of your work in problems [6] and [8] should appear as shown below, as it did earlier in Gauss/Jordan. For example, if m=1, n=2, and k=4, we write : r2 + 4 r1 = R2
 old matrix
row op name
 new matrix

[7]
[8]
[9]
Minimize w = 8x1 + 6x2
subject to:
 2x1 + 5x2 2 x1 + 2x2 3
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

[7] See alternate sample test as example of what [7][8][9] on your paper test will look like.
Here (for 5 points), you must only enter the numbers of the SIMPLEX TABLEAU, as in [5] above.
Scoring : bottom row worth 3 points, each other row worth 1 point
ST =
 [8] (6 points) Use SPECIFIC STEPS OF SIMPLEX METHOD OUTLINED IN OUR CLASS HANDOUTS, to solve the problem in [7] above. Enter numbers in decimal form to complete the needed 6 row operations below. With correct work, you will encounter 2 pivots (3 row operations in each), both in phase I of the simplex method, at which point you will arrive at a FINAL TABLEAU. All fractions in your matrices will have denominators 2,5, or 10: no others; thus, decimals will terminate after one decimal place. Scoring : You must earn a perfect 5 points in [7] above before earning any points in [8]. One point for 1st row op ; three points for the first 3 row ops ; 4 points for the first 4 row ops ; 6 points for all row ops..
On tests given in class, this highlighted component will be omitted. Both here and in class, the test-taker must also write the matrices, since without them, the 6 needed row operations ( 6 answer boxes below) cannot be discovered.
 startingtableaufrom [7]
r1 = R1
 matrix
r2 + r1 = R2

r3 +
r1 = R3
 middletableau
r2 = R2
 matrix
r1 + r2 = R1

r3 +
r2 = R3
 finaltableau
 [9] (4 pts) You encountered 3 simplex tableaus in [8] above (starting, middle, and final), not counting intermediate matrices. The basic solution for the starting tableau is :  x1 = x2 = 0, s1 = -2, s2 = -3, z = w = 0. Name the values of each of these variables for the middle and final tableaus: see diagram below. Scoring : 2 points for each fully correct basic solution

 Starting basic solution : x1 = 0 ;   x2 = 0 s1 = -2 ;   s2 = -3 z = w = 0
1st pivoting
 Middle basic solution : x1 =   ;   x2 = s1 =   ;   s2 = z = -w =
2nd pivoting
 Final basic solution : x1 =   ;   x2 = s1 =   ;   s2 = z = -w =