Introduction:
This activity requires you to use linear programming to model the constraints Brendan has for his shop and to make recommendations so that he can maximise his income in the current year and in future years. 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.
Task:
Brendan owns a shop and sells milk and toy cars. Use the information below to write a report for Brendan making recommendations as to how many boxes of each type he should sell.
Information:
 Milk require 10 hours of labour per box sold and toy cars require 4 hours of labour per box sold.
 Milk require 12 m^{2} of storage space per box sold and toy cars require 2 m^{2} of storage space per box sold.
 Brendan has a maximum of 680 hours of labour and 648 m^{2} of storage space.
 Brendan must sell at least 6 boxes of toy cars and at least 20 boxes of milk.
 For the current year, he expects to receive $1908 per box sold for his toy cars and $1340 per box sold for his milk.
 He is unsure of the future income from toy cars, but he thinks that next year will be approximately $536 per box sold.
Answer:
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Definitions:
t = boxes of toy cars
m = boxes of milk
Equations:

10m + 4t ≤ 680
12m + 2t ≤ 648
t ≥ 6
m ≥ 20
Profit (this year) = 1340 m + 1908 t
Profit (future years) = 1340 m + 536 t
Critical Points:

(20, 6)
(20, 120)
(44, 60)
(53, 6)
Milk  Toy cars  This Year  Future Years 

20  6  $38,248  $30,016 
20  120  $255,760  $91,120 
44  60  $173,440  $91,120 
53  6  $82,468  $74,236 
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