Mathematical Reasoning Question& Answer

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Question:

A company produces two types of products, A and B. The profit for each product A is $5, and the profit for each product B is $7. The company can produce a maximum of 40 units of product A and 30 units of product B per week due to resource constraints. Additionally, the total number of products produced cannot exceed 60 units per week.

Formulate a linear programming problem to maximize the weekly profit and solve it using the graphical method. Choose the correct maximum profit from the options provided.

Options:
A) $340
B) $360
C) $210
D) $200

View Answer

Let ( x ) be the number of units of product A produced per week, and ( y ) be the number of units of product B produced per week. The objective is to maximize the weekly profit ( P ), which can be expressed as:

[ P = 5x + 7y ]

Subject to the following constraints:

  1. The production of product A should not exceed 40 units:
    [ x \leq 40 ]
  2. The production of product B should not exceed 30 units:
    [ y \leq 30 ]
  3. The total production of products A and B should not exceed 60 units:
    [ x + y \leq 60 ]
  4. Non-negativity constraints:
    [ x \geq 0 ]
    [ y \geq 0 ]

To solve this using the graphical method, we first plot the constraints on a graph:

  1. ( x \leq 40 )
  2. ( y \leq 30 )
  3. ( x + y \leq 60 )

Next, we identify the feasible region defined by these constraints. The feasible region is a polygon bounded by the lines ( x = 40 ), ( y = 30 ), ( x + y = 60 ), ( x = 0 ), and ( y = 0 ).

The vertices of the feasible region are determined by the intersection points of the constraint lines. These vertices are:

  • ( (0, 0) )
  • ( (0, 30) )
  • ( (30, 30) )
  • ( (40, 20) )
  • ( (40, 0) )

Now, we calculate the value of the objective function ( P ) at each vertex:

  1. At ( (0, 0) ):
    [ P = 5(0) + 7(0) = 0 ]
  2. At ( (0, 30) ):
    [ P = 5(0) + 7(30) = 210 ]
  3. At ( (30, 30) ):
    [ P = 5(30) + 7(30) = 150 + 210 = 360 ]
  4. At ( (40, 20) ):
    [ P = 5(40) + 7(20) = 200 + 140 = 340 ]
  5. At ( (40, 0) ):
    [ P = 5(40) + 7(0) = 200 ]

The maximum value of ( P ) occurs at the vertex ( (30, 30) ), giving a maximum profit of ( 360 ) dollars.

Therefore, the company should produce 30 units of product A and 30 units of product B to maximize the weekly profit, resulting in a maximum profit of $360.

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