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      Question

      A builder needs to transport sand and gravel. A truck

      can carry 4 tons of sand and 3 tons of gravel. A van can carry 2 tons of sand and 2 tons of gravel. The site needs at least 16 tons of sand and 12 tons of gravel. If the cost per trip for a truck is ₹800 and van is ₹600, minimize the transportation cost.
      A ₹2800 Correct Answer Incorrect Answer
      B ₹3600 Correct Answer Incorrect Answer
      C ₹3200 Correct Answer Incorrect Answer
      D ₹3000 Correct Answer Incorrect Answer

      Solution

      Each truck carries 4 tons of sand and 3 tons of gravel. Each van carries 2 tons of sand and 2 tons of gravel. The requirement is:

      • At least 16 tons of sand: 4x + 2y ≥ 16
      • At least 12 tons of gravel: 3x + 2y ≥ 12
      • Non-negativity: x ≥ 0, y ≥ 0
      Objective: Minimize cost C = 800x + 600y Solve the constraints as equations (1) 4x + 2y = 16 ⇒ 2x + y = 8 (2) 3x + 2y = 12 From (1): y = 8 − 2x Substitute into (2): 3x + 2(8 − 2x) = 12 3x + 16 − 4x = 12 ⇒ −x = −4 ⇒ x = 4 Then y = 8 − 2(4) = 0 One point: (4, 0) Try other feasible combinations Try x = 2: Sand: 4×2 + 2y ≥ 16 ⇒ 8 + 2y ≥ 16 ⇒ y ≥ 4 Gravel: 3×2 + 2y ≥ 12 ⇒ 6 + 2y ≥ 12 ⇒ y ≥ 3
      So y = 4 works ⇒ Point: (2, 4) Try x = 0: Sand: 4×0 + 2y ≥ 16 ⇒ y ≥ 8 Gravel: 3×0 + 2y ≥ 12 ⇒ y ≥ 6 y = 8 satisfies ⇒ Point: (0, 8) Calculate cost at feasible points
      • At (4, 0): C = 800×4 + 600×0 = ₹3200
      • At (2, 4): C = 800×2 + 600×4 = ₹4000
      • At (0, 8): C = 800×0 + 600×8 = ₹4800
      Minimum cost = ₹3200 when 4 truck trips and 0 van trips are used.

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