A sport boutique sells two types of golf balls: standard quality at $2 each and high quality at $3.50 each. The shopkeeper notices that on a monthly basis he sells at least 2 times more standard quality balls than high quality ones. The shopkeeper

1
MTH-5101 Pretest B

• A sport boutique sells two types of golf balls
  standard quality at 2 each and high quality at
  3.50 each. The shopkeeper notices that on a
  monthly basis he sells at least 2 times more
  standard quality balls than high quality ones.
  The shopkeeper ensures that he always has at
  least 50 standard quality and 25 high quality
  balls in stack.. However, he never has more than
  150 in total. How many balls of each kind must
  he sell to maximize his revenues?
• a) Transcribe the elements needed to establish
  the constraints.
• b) Transcribe the elements needed to establish
  the function to be optimized.

• Yves is the owner of an arena. He wants to
  improve the lighting on the skating rink. To do
  the work, he hires an electrician and an
  apprentice. He evaluates the time needed to do
  the work at a maximum of 50 hours. The
  apprentice must work at least 12 hours and the
  electrician must work at least as many hours as
  the apprentice.
• The material required to complete the job costs
  10500. The electrician is paid 30 an hour and
  the apprentice is paid 18 an hour.
• a) Identify the variables.
• b) Translate the constraints into a system of
  inequalities.
• c) Express the function to be optimized.

2
3. Draw the polygon of constraints associated
with the following system of inequalities.
C1 x 0
C2 y 0
C3 x 2
C4 y 3
C5 5x - 8y 20
C6 7x 6y 150
4. Verify algebraically whether the point
(25,25) belongs to the polygon of constraints.
Show all steps to your solution.
x 0
y 0
x 20
x y 50
y 10
6x 12y 300
3
5. Determine algebraically the coordinates of
the vertices of the polygon of constraints (ABCD).
C1 x 0
C3
C2 y 0
C3 x y 450
C4 x y 150
C5 x 2y
C5
C
C4
B
D
A
4
6. A firm puts both a shampoo and a conditioner
on the market. Both products are sold in 500-ml
bottles. A study determined the following
information each month the number bottles of
shampoo sold will be at least as many as that of
the conditioner. The firm will sell a total of
at most 500 bottles in a month and at least 1500
of those will be shampoo. After a market survey,
the shampoo was priced at 3.80 per bottle and
the conditioner at 3.10 per bottle. How many
bottles of each kind must the firm sell to
maximize its revenue? What would be that
revenue? Show all of your work.
5
7. John makes 2 types of badges sport badges
and ecological badges. The sport badges cost 8
each to make whereas the ecological badges cost
13. John cant produce more than 90 badges. He
must produce at least 10 sport badges and 20
ecological badges. He must produce at least as
many ecological badges as sport badges. His
advertising campaign costs 100. How many badges
of each type must he produce to minimize his
costs. What would be that cost? Show all of
your work.
6

• At the district playground, the instructors
  planned an outdoor activity at the Bromont
  slides. Buses had to be rented to transport
  between 120 and 240 children. Two kinds of buses
  were available 15-passenger and 30-passenger.
  Each small bus cost 40 and each big bus cost50.
• The polygon of constraints is presented in the
  graph on the right where

x the number of 15-passenger buses y the
number of 30-passenger buses
The instructors later learn that they must rent
at least 4 30-passenger buses. How will this
effect the decision on minimizing the cost of the
outdoor activity.
8
6
4
2
4
8
16
12