Q10 of 31 Page 519

There are two types of fertilisers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1 costs Rs 6/kg and F2 costs Rs 5/kg, determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

let the farmer buy x kf of fertilizer f1 and y kg of fertilizer f2

∴ x and y ≥ 0


The tabular representation of the data is as follows:



F1 consists of 10% nitrogen and F2 consists of 5% nitrogen. But the farmer requires at least 14 kg of nitrogen.


10% × x + 5% × y ≥ 14



2x+y ≥ 280


food F1 consists of 6% phosphoric acid and food F2 consists of 10% phosphoric acid. But the farmer requires at least 14kg of phosphoric acid


⇒ 6% × x + 10% of y ≥ 14


⇒


3x + 56y ≥ 700


Total cost of fertilizers, Z= 6x+ 5y


The mathematical formulation of the given data is


Minimize, Z = 6x +5y


Subject to constraints


2x+y ≥ 280


3x + 56y ≥ 700


x and y ≥ 0


The feasible region determined by the system of constraint is:



It can be seen from the graph that the feasible region is unbounded


The corner points are


The value of Z at these points are:



As the feasible region is unbounded, 1000 may or may not be the minimum value


For this we will draw graph of inequality


6x+ 5y < 1000


It can be seen there is no common point between feasible region and 6x+ 5y < 1000


∴ 100 KG of fertilizer F1 and 80 kg of fertilizerF2 should be used to minimize the cost and the minimum cost is Rs 1000.


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8

A merchant plans to sell two types of personal computers – a desktop model and a portable model that will cost Rs 25000 and Rs 40000 respectively. He estimates that the total monthly demand of computers will not exceed 250 units. Determine the number of units of each type of computers which the merchant should stock to get maximum profit if he does not want to invest more than Rs 70 lakhs and if his profit on the desktop model is Rs 4500 and on portable model is Rs 5000.

 9

A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1 costs Rs 4 per unit food and F2 costs Rs 6 per unit. One unit of food F1 contains 3 units of vitamin A and 4 units of minerals. One unit of food F2 contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.

 11

The corner points of the feasible region determined by the following system of linear inequalities:

2x + y ≤ 10, x + 3y ≤ 15, x, y ≥ 0 are (0, 0), (5, 0), (3, 4) and (0, 5). Let


Z = px + qy, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both (3, 4) and (0, 5) is


A. p =q B. p = 2q


C. p = 3q D. q = 3p

 1

Refer to Example 9. How many packets of each food should be used to maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet?