Achievement Standard Mathematics and Statistics 91574:
Apply linear programming methods in solving problems
Resource reference: Mathematics and Statistics 3.2A
Resource title: Pizza Shack
Credits: 3
Achievement
Apply linear programming methods
in solving problems.
Achievement with Merit
Achievement with Excellence
Apply linear programming methods,
using relational thinking, in solving
problems.
Apply linear programming methods,
using extended abstract thinking, in
solving problems.
Introduction
This activity requires you to use linear programming to investigate a situation that involves maximising a function in
two variables subject to various constraints. You will present your findings as a written report, supported by graphs,
equations and relevant calculations. The quality of your reasoning and how well you link this to the context will
determine the overall grade.
Context
Pizza Shack is a restaurant that sells pizza, amongst other things. They sell two types of pizza: Value pizzas and
Gourmet pizzas.

Value pizzas cost $3.50 to produce and 2 minutes for one person to prepare.

Gourmet pizzas cost $5.00 to produce and 4 minutes for one person to prepare.

Value pizzas are more popular than Gourmet pizzas. From experience, the manager knows that Pizza Shack
sells at least twice as many Value pizzas as Gourmet pizzas on any given night.

Pizza Shack is open from 4:30pm to 10:00pm each night. During this time two people work on preparing
pizzas.

The manager has budgeted $1000 per day for pizza production.
Let x be the number of Value pizzas made and y be the number of Gourmet pizzas made.
Task
This activity requires you to use linear programming to model the constraints Pizza Shack have to consider when
making pizzas and to answer questions for the following scenarios.
You will present your findings as a written report, supported by graphs, equations and relevant calculations. As you
write your report take care to clearly communicate your findings using appropriate mathematical statements and show
mathematical insight in relating your answers to the context.
SCENARIO ONE
Currently Value pizzas sell for $5.00 and Gourmet pizzas sell for $8.50. Determine how many pizzas of each type
need to be sold per day to maximise the manager’s revenue, and clearly state this revenue.
SCENARIO TWO
The manager considers selling Gourmet pizzas for a different price. Discuss the price that he should sell the Gourmet
pizzas for so that there is more than one way to generate a maximum revenue from pizza sales?
Assessment schedule: Mathematics and Statistics 91574 Pizza Shack
Evidence/Judgements for Achievement
Evidence/Judgements for Achievement with Merit
The student has applied linear programming methods
in solving problems to make a recommendation about
the optimum number of birds.
The student has applied linear programming methods,
using relational thinking, in solving problems to make
a recommendation about the optimum number of
birds.
The student has linear programming methods, using
extended abstract thinking, in solving problems to
make a recommendation about the optimum number
of birds and the effect on profit.
The student has demonstrated relational thinking by
 forming and using a model AND
 relating their findings to the context, or
communicating their thinking using appropriate
mathematical statements
The student has demonstrated extended abstract
thinking by
 devising a strategy to investigate or solve a
problem AND
 using correct mathematical statements or
communicating mathematical insight.
The student has demonstrated this by



selecting and using methods
demonstrating knowledge of concepts and terms
communicating using appropriate
representations.
For example:
 Finding the equation of a linear inequality, for
example,
Cost: 3.5x + 5y < 1000
Time: 2x + 4y < 660
Ratio: x ≥ 2y
Non negativity: x ≥ 0, y ≥ 0

For example:
The student correctly graphs the linear system and
identifies the feasible region. They have
recommended the optimal number pizzas maximise
the profit.
Representing the feasible region graphically,
correctly showing at least two of the constraints.
Investigate integer values
See graph below
(175, 78) Not in the region

Give potential vertices that maximises profit for the
feasible region OR gradient method
(174,78) $1533

Objective Function:

R = 5x + 8.5y

Vertices: (175, 77.5) $1533.75

(165, 82.5) $1526.25
(175, 77) $1529.50
This is close to (165, 82) $1522.00
(165, 83) not in the region
Answer in context and explained or calculations or
objective function gradient used.
Value pizzas 174, Gourmet pizzas 78 for maximum
profit of $1533
The manager is unlikely to make this profit as it would
mean the staff would have to work without a break.
Evidence/Judgements for Achievement with
Excellence
For example:
The student has formed and used a system of linear
constraints to find the optimal solution.
Students need to say integer solutions that lie on the
line between the two points.
There are two options:
If objective function has the same gradient as the
cost equation:
-3.5 -5
=
Þ A = $7.14
5
A
time constraint:
-2 -5
=
Þ A = $10
4
A
The manager could use either to give multiple
solutions. He should round the $7.14 to $7.10 if he
wishes to keep it cheaper or choose $10 if he wishes
to make it more expensive. A change in price might
change the buying habits of customers..
Final grades will be decided using professional judgement based on a holistic examination of the evidence provided against the criteria in the
Achievement Standard.