Mathematics question for Higher National Diploma (HND) in Cameroon.

A farmer has a rectangular field that is 60 meters long and 40 meters wide. He wants to fence the entire field with wire. If the wire costs $2 per meter, how much will it cost the farmer to fence the field?

Question:

A manufacturer produces two types of products, A and B. The profit on each unit of product A is 10,000 FCFA and the profit on each unit of product B is 15,000 FCFA. The manufacturer has a production capacity of 100 units per day. The demand for product A is limited to 40 units per day, and the demand for product B is unlimited.

a) Formulate a linear programming model to maximize the daily profit.
b) Solve the model using the graphical method.
c) What is the optimal production plan and the maximum daily profit?

Answer:

a) Linear Programming Model:

Objective function: Maximize profit = 10,000A + 15,000B

Constraints:
- Production capacity: A + B ≤ 100
- Demand for product A: A ≤ 40
- Non-negativity: A ≥ 0, B ≥ 0

b) Graphical Method:

Plot the constraints and find the feasible region.

```
[Image of constraints and feasible region]
```

From the feasible region, identify the corner points: (0, 6.67), (40, 0), (0, 100), and (80, 20).

Evaluate the objective function at each corner point:

```
(0, 6.67): 10,000(0) + 15,000(6.67) = 100,050
(40, 0): 10,000(40) + 15,000(0) = 400,000
(0, 100): 10,000(0) + 15,000(100) = 1,500,000
(80, 20): 10,000(80) + 15,000(20) = 1,400,000
```

c) Optimal Production Plan and Maximum Daily Profit:

The maximum daily profit of 1,500,000 FCFA is achieved by producing 80 units of product A and 20 units of product B.