An agronomist needs to supply exactly of elemental nitrogen to a field while keeping…

MATHEMATICS · Multi-part · CBSE · Mathematics · Linear Programming · Linear Programming

Problem

An agronomist needs to supply exactly

150\,\text{kg}

of elemental nitrogen

(\mathrm{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{dollars kg}^{-1}

. Fertilizer B:

30\%

N by mass, price

=0.25\,\text{dollars kg}^{-1}

. Fertilizer C:

15\%

N by mass, price

=0.14\,\text{dollars kg}^{-1}

. Additional constraints: Total mass applied to the field must not exceed

500\,\text{kg}

. For micronutrient supplementation, at least

100\,\text{kg}

of Fertilizer C must be included. (1) (a) Define variables for the masses of the three fertilizers and write the complete set of algebraic constraints that these variables must satisfy. (2) (b) Express the total cost of the fertilizer blend solely in terms of the defined variables. (3) (c) Using algebraic reasoning, determine the combination of fertilizers that meets all constraints and yields the minimum possible cost. Show all steps clearly. (4) (d) State the masses of Fertilizers A, B, and C to be applied and the corresponding total cost.

Related theory

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