Linear Programming, Gaussian Elimination and Matrix

Extracted Content from Question Files:

PART B. SHORT ANSWER (Problems #8-12, 65 points total).

8. (6 pts) There is a jar containing nickels and quarters.
Suppose there are 89 coins in the jar, and the total value is $19.45.
We would like to find out how many nickels and how many quarters are in the jar.

Let x = number of nickels and y = number of quarters.

Write a system of two linear equations involving variables x and y, corresponding to the information
given about the coins. [Just set up the system of linear equations. You are not asked to solve the system.]
9. (20 pts)
Consider the system of three linear equations: 0 0 0 0 
0 0 0 0 
Template:
 
 x − y + 2z = 3
0 0 0 0 
  
 − x + 2 y − 3 z = −2 (If you wish, you can use the Equation Editor template
 2x − 2 y + z = 3
 shown above, to make the typing easier. Just copy, paste
where appropriate, and edit with the Equation Editor as needed.)

(a) Write the augmented 3 x 4 matrix corresponding to this system.

(b) Solve the system by using the Gaussian elimination method. Show work. (When you are done, it is a
very good idea to check your solution by substitution into the original system of equations, but do not go to the trouble
of submitting your check.)
10. (6 pts) Let U = {1, 2, 3, 4, 5, 6, 7}, A = {1, 3, 5, 6} and B = {1, 2, 6}.
List the elements of the indicated sets.

(a) A ∩ B

(b) A′ ∪ B
(Be sure to notice the complement symbol applied to A)

11. (8 pts) In a certain state legislature of 120 legislators, there are 60 Democrats, 40 Republicans,
and 20 Independents.
Republicans
Democrats Independents
L = [60 20]
40

The following matrix summarizes (in decimal format) the percentage of legislators who voted for or
against a particular finance bill, by affiliation.
For Against
 0 .6 0.4 Democrat
P =  0 .3 0.7  Republican
 
 1 .0 0 Independent
 

For instance, 70% of Republican legislators voted against the finance bill.

(a) Determine a matrix calculation resulting in a matrix having two entries, where the first entry is
the vote total for the bill and the second entry is the vote total against the bill.
State the matrix calculation and the state the entries in the resulting matrix. (For example, a matrix
calculation for matrices A and B might be A + B or A – B or AB or BA, etc. What is the matrix calculation involving L
and P in this problem? Show the steps in the calculation and the result.)

(b) A bill passes if a majority (over half) of the legislators vote for the bill. By looking at your
results from (a), did the bill pass or not?
12. (25 pts) Two kinds of cargo, A and B, are to be shipped by truck. Each crate of cargo A is 50
cubic feet in volume and weighs 200 pounds, whereas each crate of cargo B is 10 cubic feet in
volume and weighs 360 pounds. The shipping company earns $80 per crate for cargo A and $100
per crate for cargo B. The truck has a maximum load limit of 1,000 cubic feet and 7,200 pounds.
The shipping company would like to earn the highest revenue possible.

(a) Fill in the chart below as appropriate.

Cargo A Cargo B Truck Load Limit
(per crate) (per crate)

Volume

Weight

Revenue

(b) Let x be the number of crates of cargo A and y the number of crates of cargo B shipped by one
truck. Using the chart in (a), give two inequalities that x and y must satisfy because of the truck’s
load limits.

(c) Give two inequalities that x and y must satisfy because they cannot be negative.

(d) Give an expression for the total revenue earned from shipping x crates of cargo A and y crates of
cargo B.

(e) State the linear programming problem which corresponds to the situation described. That is, be
sure to indicate whether you have a maximization problem or a minimization problem, and state the
objective function and the constraints. (This part is mostly a summary of the previous parts)
(f) List the inequalities from part (e) in standard form.

(g) Solve the linear programming problem. You will need to find the feasible set and determine the
vertices. You do not have to submit your graph, and you do not have to show your algebraic work
in finding the vertices, but you must list your vertices and corresponding values of the objective
function.

Vertex (x, y) Objective Function

(h) Write your conclusion with regard to the word problem. State how many crates of cargo A and
how many crates of cargo B should be shipped in one truck, in order to earn the highest total
revenue possible. State the value of that maximum revenue.

MATH 106 QUIZ 2 – MULTIPLE CHOICE PORTION February-March, 2009

MULTIPLE CHOICE (35 points total, 5 points per problem). For problems #1-7, choose the best
answer. Record your choices on the answer sheet provided with the short answer quiz
document. You are NOT required to show work for the multiple choice problems .

1. (5 pts) The result of performing the elementary row operation [3] + (5)[2] 1. _______
1 3 9
0
0 1 − 3 2 is
on the system  
0 − 5 4 1
 

1 0 3 9  1 0 3 9 1 0 3 9 1 3 9
0
0 1 − 3 2  B. 0 1 −3 2   0 1 − 3 2 0 − 5 − 11 11
A.  C.  D. 
  
 

0 0 − 11 11 
0 0 4 1 10 5 14 11  4 1
0 − 5
    

2. (5 pts) When solving a system of linear equations with the unknowns x , y, and z 2. _______
using the Gaussian elimination method, the following augmented matrix was obtained.

1 0 0 3
 
0 1 2 4 What can be concluded about the solution of the system?

0 0 0 1


A. There is exactly one solution.
B. There are no solutions.
C. There are infinitely many solutions.
D. The number of solutions cannot be determined.

− 1 6  − 1 3
− 
3. (5 pts) Calculate the matrix subtraction.   3. _______
 0 2   6 1

0 − 3 − 2 9 
 0 3  3
A.   B.   C.   D.  
− 6 1 6 − 1 − 5  6 3
 2 1  5 4 0
4. (5 pts) Find the matrix product  4. _______
 
− 1 0  − 6 1 7

10 4   10 8 0 4 9 7  16 7 − 7
A.  B.  C.  D. 
   4 0
 6 −1 7 − 5 −4 0
 6 0 5 

(Coffee Merchant) A coffee merchant sells two blends of coffee.
For each pound of Blend A, 80% is Mocha Java and 20% is Jamaican, and Blend A sells for $2.10 a pound.
For each pound of Blend B, 30% is Mocha Java and 70% is Jamaican, and Blend B sells for $2.85 a pound.
The merchant has available 1000 pounds of Mocha Java and 600 pounds of Jamaican.
The merchant will try to sell the amount of each blend that maximizes her income.
Let x be the number of pounds of Blend A and let y be the number of pounds of Blend B.

5. (5 pts) For the Coffee Merchant situation described above, state the objective function.
5. _______

A. 1000x + 600y

B. 0.30x + 2.10y

C. 0.80x + 0.20y

D. 2.10x + 2.85y

6. (5 pts) For the Coffee Merchant situation described above, determine which of the following
inequalities must be satisfied, regarding Mocha Java. 6. _______

0.80x + 0.20y ≤ 1000
A.

0.80x + 0.30y ≤ 1000
B.

0.30x + 0.70y ≥ 1000
C.

0.80x + 0.30y ≥ 1000
D.
7. (5 pts) 7. _______

Given the feasible set shown to the right, find
the values of x and y that minimize the
objective function 5x + 6y.

A. (x , y) = (0, 7)

B. (x , y) = (1, 4)

C. (x , y) = (3, 2)

D. (x , y) = (6, 0)

End of the multiple choice portion. Go to the short answer document to record your answer
choices for submission.