A furniture manufacturer produces two types of display cabinets, type X and type Y. On a

weekly basis he must produce at least 2 of each type, but not more than 5 of type X or more

than 6 of type Y. It takes 4 hours to produce type X and 5 hours for type Y in a 40 hour

working week. At least 12 workers are needed with 2 working on type X and 3 on type Y at

any one time.

2.1 Represent the above information as a system of inequalities .

2.2 Draw a graph of the system and indicate the feasible region clearly.

2.3 If the profit (P) on type X is R800 and on type Y is R1000, write down the objective

function in the form P = ax + by .

2.4 Determine the number of each type that must be produced each week to make a

maximum profit. Determine the maximum profit

 

Answer:
Let

?x= the number of units of cabinet type X?

?y= the  number of untis of cabinet type Y?

Since our goal is to maximize profit, the object function is given by ( See the table above for a summary of how the constraint inequalities were obtained.

2.3

?Max  P=800x+1000y?

2.1

subject to

?4x+5y≤40?

?2x+3y≥12?

?2≤x≤5?

?2≤y≤6?

?x , y≥0?

Sketch the graph of each of the linear equations obtained converting the constraint inequalities to equality

?1.     4x+5y=40     ?y=540−4x?=8−0.8x?

?2.    2x+3y=12     ?y=312−2x?=4−32?x?

2.4

The graph of the inequalities shows the the maximum profit will occur at point D(5,4)and the maximum profit is given by

?P(5,4)=800⋅5+1000⋅4=R 8000?

Note the point F(2.5,6) does not give maximum profit since a half cabinet can not be sold. The profit this point is R 8000 (P(2,6) round down 2.5 to 2).

2.2

please use this google drive link to download the answer file.

https://drive.google.com/file/d/1MeOVGIOvz6ZQRsnVQgvarSonTiUKvTYD/view?usp=sharing

note: if you have any trouble in viewing/downloading the answer from the given link, please use this below guide to understand the whole process.

https://helpinhomework.org/blog/how-to-obtain-answer-through-google-drive-link