Question

The maximum value of Z= 2x + 3y subject to the constraints x≥0, y>0; x+y≤ 10, 3x+4y≤ 36 is:

• A. 20
• B. 27
• C. 30
• D. 0

Answer 1

The Correct Option is B

To determine the maximum value of Z=2x+3y given the constraints x≥0, y>0; x+y≤10, and 3x+4y≤36, we will use the graphical method for solving linear programming problems.

Step	Description
1	Identify the feasible region by graphing the constraints on the xy-plane.
2	Graph x+y=10. This is a line passing through the points (0,10) and (10,0).
3	Graph 3x+4y=36. This line passes through (0,9) and (12,0).
4	Identify the feasible region where all constraints overlap, considering x≥0 and y>0.
5	Determine the vertices of the feasible region. They are (0,0), (0,9), (4,6), and (10,0).
6	Calculate Z=2x+3y for each vertex: (0,0): Z=2(0)+3(0)=0 (0,9): Z=2(0)+3(9)=27 (4,6): Z=2(4)+3(6)=8+18=26 (10,0): Z=2(10)+3(0)=20
7	Identify the maximum value: Z=27 at (0,9).

The maximum value of Z is 27, confirming the correct answer is 27.