Worksheet Chapter 7: Linear Programming

1. An event organizer plans to buy x wooden chairs and y plastic chairs based on the following constraints: 1. The total number of chairs is not more than 100. 2. The number of plastic chairs is at most 3 times the number of wooden chairs. 3. The number of plastic chairs exceeds wooden chairs by at least 10. (a) Write three inequalities representing these constraints. (b) Using a scale of 2 cm to 20 chairs on both axes, identify the vertices of the feasible region R. (c) Find the range of the number of wooden chairs if 50 plastic chairs are purchased.

2. A transport company uses x lorries and y vans to deliver goods. The cost of operating a lorry is RM150 per trip and a van is RM100 per trip. The constraints are: - x + y ≤ 12 - y ≥ 2 - 2x + y ≥ 8 Find the combination of (x, y) that minimizes the total operating cost C = 150x + 100y, and calculate this minimum cost.

3. A farmer plants x hectares of corn and y hectares of soybeans. The profit function is P = 400x + 350y (in RM). The feasible region has vertices at (0, 0), (0, 40), (20, 30), and (35, 0). (a) Calculate the profit at each vertex. (b) State the maximum profit and the corresponding allocation of land.

4. A tailor sews x shirts and y trousers. The cost to produce a shirt is RM20 and a trouser is RM30. The total budget available is RM600. Furthermore, the number of trousers sewn must be at least 5. (a) Write two inequalities involving x and y. (b) If the tailor makes 12 shirts, find the maximum number of trousers that can be sewn within budget.

5. Which of the following points lies in the feasible region bounded by x ≥ 0, y ≥ 0, x + 2y ≤ 10, and y ≤ x + 1?

6. State whether the statement is True or False: 'The point (3, 5) satisfies the inequality 2x + y > 12.'

7. Fill in the blank: The region that satisfies all the given linear constraints simultaneously in linear programming is called the ________ region.

8. A tuition center offers two courses: A (x students) and B (y students). The constraints are: 1. x + y ≤ 120 2. y ≥ x2 3. x ≤ 70 If the fee for Course A is RM80 and Course B is RM60 per student: (a) Write the objective function for total revenue R. (b) Identify all vertices of the feasible region. (c) Determine the maximum total revenue collected by the tuition center.

9. A contractor constructs x single-storey houses and y double-storey houses under the following conditions: I: The total number of houses to be built is at most 80. II: The number of double-storey houses is at least half the number of single-storey houses. III: The number of double-storey houses exceeds single-storey houses by not more than 20. Write down three linear inequalities, other than x ≥ 0 and y ≥ 0, which satisfy all the above conditions.

10. Write down the linear inequality for the statement: 'The total number of model A cars (x) and model B cars (y) produced per day is at least 30.'