STUDY GUIDE: LINEAR PROGRAMMING
1. A steel company produces two types of machine dies, part A and part B. The company makes a $2.00 profit
on each part A that it produces and a $5.00 profit on each part B that it produces. Let x = the number of part
A produced in a week and y = the number of part B produced in a week.
Write the objective function that describes the total weekly profit.
a. z = 2x + 5y
b. z= =2(x - 5) + 5(y - 2)
c. z = 5x + 2y
d. z = 7(x + y)
2. A dietitian needs to purchase food for patients. She can purchase an ounce of chicken for $0.25 and an
ounce of potatoes for $0.02. The dietician is bound by the following constraints.
∙ Each ounce of chicken contains 13 grams of protein and 24 grams of carbohydrates.
∙ Each ounce of potatoes contains 5 grams of protein and 35 grams of carbohydrates.
∙ The minimum daily requirements for the patients under the dietitian's care are 45 grams of protein and
58 grams of carbohydrates.
Let x = the number of ounces of chicken and y = the number of ounces of potatoes purchased per patient.
Write a system of inequalities that describes these constraints.
A.
B.
C.
D.
3. Mrs. White wants to crochet hats and afghans for a church fundraising bazaar. She needs 6 hours to make a
hat and 4 hours to make an afghan, and she has no more than 56 hours available. She has material for no
more than 12 items, and she wants to make at least two afghans. Let x = the number of hats she makes and
y = the number of afghans she makes. Write a system of inequalities that describes these constraints.
A)
B)
C)
D)
6x + 4y ≤ 56
6x + 4y ≤ 56
6x + 4y ≥ 56
4x + 6y ≤ 56
x + y ≤ 12
x + y ≤ 12
x + y ≤ 12
x + y ≤ 12
y≥2
x≥2
y≥2
x≤2
4. Find the maximum or minimum value of the given objective function of a linear programming problem.
The figure illustrates the graph of feasible points.
Objective Function: z = 5x + 6y.
a) 40
b) 32
Find minimum.
c) 33
d) No minimum
#5-6: An objective function and a system of linear inequalities representing constraints are given. Graph
the system of inequalities representing the constraints. Find the value of the objective function at each
corner of the graphed region. Use these values to determine the maximum value of the objective function
and the values of x and y for which the maximum occurs.
5.
Objective Function
Constraints
z = 15x + 9y
0 ≤ x ≤ 10
0≤y≤5
3x + 2y ≥ 6
A. Maximum: 27; at (0, 3)
C. Maximum: 45; at (0, 5)
6.
Objective Function
Constraints
a. Maximum 60; at (6, 3)
c. Maximum 66; at (6, 4)
7.
B. Maximum: 45; at (0, 5)
D. Maximum: 150; at (10, 0)
z = 7x + 6y
x≥0 y≥0
3x + y ≤ 21
x + y ≤ 10
x + 2y ≥ 12
b. Maximum 68; at (8, 2)
d. Maximum 65.5; at (5.5, 4.5)
A company manufactures two products X and Y. Each product has to be processed in three
departments: welding, assembly and painting. Each unit of X spends 2 hours in the welding department,
3 hours in assembly and 1 hour in painting. The corresponding times for a unit of Y are 3, 2 and 1 hours
respectively. The employee hours available in a month are 1,500 for the welding department, 1,500 in
assembly and 550 in painting. The contribution to profits are $100 for product X and $120 for product
Y.
A) What is the objective function (Z)
B) What is the equation of the labor
to be maximized in this linear
constraint line for the welding
programming problem, where Z is
department?
total profit in $)
a. 2x + 3y = 550 hours
a. Z = 2x + 3y
b. 3x + 2y = 1500 hours
b. Z = 1500x + 1500y
c. 3x + 2y = 550 hours
c. Z = 100x +120y
d. 2x + 3y = 1500 hours
d. Z = 120x + 100y
C) What is the equations of the labor
constraint line for the assembly
department?
a. 3x + 2y = 1500 hours
b. x + y = 550 hours
c. 2x + 2y = 1500 hours
d. x + y = 1500 hour
D) What is the solution to this linear
programming problem in terms of
the respective quantities of X and Y
to be produced if profits are to be
maximized?
a. X = 150, Y = 400
b. X = 0, Y = 500
c. X=400, Y=150
d. X = 550, Y = 0
8.
David and Harry operate a discount jewelry store. They want to determine the best mix of customers to
serve each day. There are two types of customers for their store, retail (R) and wholesale (W). The cost
to serve a retail customer is $70 and the cost to serve a wholesale customer is $89. The average profit
from either kind of customer is the same. To meet headquarters' expectations, they must serve at least 8
retail customers and 12 wholesale customers daily. In addition, in order to cover their salaries, they must
at least serve 30 customers each day. Which of the following is one of the constraints for this model?
A.
B.
C.
D.
E.
1R+1W≤8
1 R + 1 W ≥ 30
8 R + 12 W ≥ 30
1 R ≥ 12
20 (R + W) ≥30
9. Apply linear programming to this problem. A one-airplane airline wants to determine the best mix of
passengers to serve each day. The airplane seats 25 people and flies 8 one-way segments per day. There
are two types of passengers: first class (F) and coach (C). The cost to serve each first class passenger is
$15 per segment and the cost to serve each coach passenger is $10 per segment. The marketing
objectives of the airplane owner are to carry at least 13 first class passenger-segments and 67 coach
passenger-segments each day. In addition, in order to break even, they must at least carry a minimum of
110 total passenger segments each day. Which of the following is one of the constraints for this linear
program?
A. 15 F + 10 C ≥ 110
1 F + 1 C ≥ 80
B.
C. 13 F + 67 C ≥ 110
D. 1 F ≥ 13
13+ 67 C ≤ (80/200)
E.
10. Leo has $12.50 to spend on his weekly supply of sweets, crisps and apples. A bag of crisps costs $0.65,
a bag of sweets costs $0.85, and one apple costs $0.50. The total number of packets of crisps, sweets and
apples consumed in a week must be at least seven, and he eats at least twice as many packets of sweets
as crisps. His new healthy diet also means that the total number of packets of sweets and crisps must not
exceed one-third of the number of apples. If s, c and a, denote the number of packets of sweets, packets
of crisps, and apples respectively, which one of the following represents one of the constraints defining
the feasible region?
A. 3c + 3s ≤ a
B. a + c + s > 7
C. s ≤ c – a
D. s ≥ 0.5c
E. 0.65s + 0.85c + 0.5a ≥ 12.5
SYSTEMS OF EQUTIONS:
Determine whether the given ordered pair is a solution of the system.
11. Use the graph to the right to determine which of the following is NOT a
solution to the system:
a) (0, 0)
b) (-3, 0)
c) (0, 5)
d) (4, 0)
12) ( 5, 6)
x + y = 11
x - y = -1
A) not a solution
B) solution
14) ( -5, 4)
3x = -11 - y
2x = 2 - 3y
A) not a solution
B) solution
Solve the system of equations by the substitution method.
15)
A) {( -2, 8)}
B) {( 2, -8)}
Solve the system by the addition method.
16) x + y = -5
x - y = 12
A) {( 3.5, 8.5)}
B) {( 5, -8.5)}
17) x + 6y = 24
-6x + 5y = -21
A) (6, 3)
B) (-6, 4)
C) {( 2, 8)}
D) {( -2, -8)}
C) {( 3.5, -8.5)}
D) {( 5, 3.5)}
C) (5, 4)
D) No Solution
18) One number is 6 less than a second number. Twice the second number is 2 less than 4 times the first. Find the two
numbers.
A) 6 and 12
B) 7 and 13
C) -13 and -7
D) 8 and 14