An agronomist needs to supply exactly 150kg of elemental nitrogen (N) to a field while keeping fertilizer costs as low as possible. Three solid products are available.
• Fertilizer A: 46\% N by mass, price = \0.45\text{kg}^{-1}$
• Fertilizer B: 30\% N by mass, price = \0.25\text{kg}^{-1}$
• Fertilizer C: 15\% N by mass, price = \0.14\text{kg}^{-1}$
Additional constraints:
– Total mass applied to the field must not exceed 500kg.
– For micronutrient supplementation, at least 100kg of Fertilizer C must be included.
a. Part 1: (a) Define variables for the masses of the three fertilizers and write the complete set of algebraic constraints that these variables must satisfy.
b. Part 2: (b) Express the total cost of the fertilizer blend solely in terms of the defined variables.
c. Part 3: (c) Using algebraic reasoning, determine the combination of fertilizers that meets all constraints and yields the minimum possible cost. Show all steps clearly.
d. Part 4: (d) State the masses of Fertilizers A, B, and C to be applied and the corresponding total cost.