Solution

The correct answer is 1) A and C only.

Key Points

• To determine the solution of the given linear programming problem (LPP), we need to check which of the given points satisfy all the constraints and provide the minimum value for the objective function.
• First, we check the constraints for each point:
  • For point (4, 0):
    • 3(4) + 4(0) ≤ 60 → 12 ≤ 60 (true)
    • 5(4) - 3(0) ≥ 20 → 20 ≥ 20 (true)
    • x, y ≥ 0 → true
  • For point (2, 0):
    • 3(2) + 4(0) ≤ 60 → 6 ≤ 60 (true)
    • 5(2) - 3(0) ≥ 20 → 10 ≥ 20 (false)
    • x, y ≥ 0 → true
  • For point (7, 5):
    • 3(7) + 4(5) ≤ 60 → 21 + 20 ≤ 60 → 41 ≤ 60 (true)
    • 5(7) - 3(5) ≥ 20 → 35 - 15 ≥ 20 → 20 ≥ 20 (true)
    • x, y ≥ 0 → true
  • For point (0, 15):
    • 3(0) + 4(15) ≤ 60 → 60 ≤ 60 (true)
    • 5(0) - 3(15) ≥ 20 → -45 ≥ 20 (false)
    • x, y ≥ 0 → true
  • For point (8, 5):
    • 3(8) + 4(5) ≤ 60 → 24 + 20 ≤ 60 → 44 ≤ 60 (true)
    • 5(8) - 3(5) ≥ 20 → 40 - 15 ≥ 20 → 25 ≥ 20 (true)
    • x, y ≥ 0 → true
• Only the points (4, 0), (7, 5), and (8, 5) satisfy all the constraints.
• Now, we calculate the objective function z = 30x - 18y for these points:
  • For point (4, 0): z = 30(4) - 18(0) = 120
  • For point (7, 5): z = 30(7) - 18(5) = 210 - 90 = 120
  • For point (8, 5): z = 30(8) - 18(5) = 240 - 90 = 150
• The minimum value of the objective function is 120, which is achieved by points (4, 0) and (7, 5).

Therefore, the correct answer is 1) A and C only.

Additional Information

• Linear programming problems involve optimizing a linear objective function subject to a set of linear constraints.
• The feasible region is the set of all points that satisfy the constraints.
• The optimal solution is a point in the feasible region that minimizes or maximizes the objective function.