Question

A person requires at least 4 kilograms of phosphate and 3 kilograms of nitrogen to fertilise a lawn. They plan to make a mixture of two different fertiliser brands. Shirley 17 costs $20a bag and yields 1 kilogram of phosphate and 0.5 kilograms of nitrogen. Grow Turf costs $25 a bag and yields 1 kilogram of phosphate and 1 kilogram of nitrogen. The aim is to find the number of bags of each kind of fertiliser that must be purchased in order to minimise the cost. The numbers of bags are not required to be integers. Formulate a linear programming model for this problem.

Answer 1

A linear programming model to minimize the cost of fertilizers requires defining variables for the bags of each brand, an objective function to minimize cost, and constraints for the phosphate and nitrogen requirements.

Step-by-step explanation:

To formulate a linear programming model for this problem, let's define two variables:

• x: the number of bags of Shirley 17 fertilizer
• y: the number of bags of Grow Turf fertilizer

The objective function represents the total cost, which we want to minimize:

Minimize Z = 20x + 25y

Subject to the following constraints based on the phosphate and nitrogen requirements:

1. x + y ≥ 4 (At least 4 kg of phosphate)
2. 0.5x + y ≥ 3 (At least 3 kg of nitrogen)
3. x ≥ 0 (Non-negativity constraint for Shirley 17)
4. y ≥ 0 (Non-negativity constraint for Grow Turf)

With the objective function and constraints set, we can use a method such as the graphical method or the simplex method to find the values of x and y that will minimize the cost while meeting the fertilizer requirements for phosphate and nitrogen.