Question

A farmer has 20 hectares for growing barley and swedes. The farmer has to decide how much of each crop to grow. The cost per hectare for barley is $30 and for swedes is $20. The farmer has budgeted $480. Barley requires 1 man-day per hectare and swedes require 2 man-days per hectare. There are 36 man-days available. The profit on barley is $100 per hectare and on swedes is $120 per hectare. Find the number of hectares of each crop the farmer should sow to maximize profits.

Answer 1

The problem at hand is a typical linear programming question in mathematics, where the student must determine the optimal allocation of hectares to barley and swedes to maximize profits, given budget and manpower constraints.

Step-by-step explanation:

The student is presented with a problem that involves allocating resources to maximize profit under certain constraints. This is a linear programming problem, which is a part of mathematics. To solve it, we need to define two variables: let x be the number of hectares for growing barley and y be the number of hectares for growing swedes. The constraints given are related to budget and available man-days, which can be written as:

• 30x + 20y ≤ 480 (Budget Constraint)
• x + 2y ≤ 36 (Man-days Constraint)
• x + y ≤ 20 (Land Constraint)
• x ≥ 0 and y ≥ 0 (Non-negativity Constraint)

The objective function, which is to maximize profit, is given by:

Profit = 100x + 120y

Using the method of linear programming, either graphically or by using a simplex algorithm, one can determine the optimal number of hectares to allocate to each crop to maximize profit while satisfying all constraints.