Question

A health food manufacturer makes two types of protein shakes, Whey and Vegan. Each mixture
contains vitamins, protein, and carbohydrates, in the following proportions:
Carbohydrates (g) Protein (g) Vitamin (g)
Whey 100 400 300
Vegan 200 300 100
Minimum dietary requirements are that a protein shake contain at least 0.9kg of vitamins, 2.2kg of
protein, and 0.8kg of carbohydrate. Whey costs N$20 per kilogram to produce, and Vegan costs
$12.50 per kilogram to produce. Find the number of kilograms of each protein shake that should be
produced to minimize cost.

Answer 1

Let the firm manufacture x kg of Whey and y kg of Vegan Protein shake respectively.

According to the given condition,

The equation for carbohydrate, protein and vitamins are respectively.

100x+200y $\ge$ 800

x+2y$\ge$ 8.....(1)

400x+300y$\ge$ 2200

4x+3y$\ge$ 22....(2)

300x+100y$\ge$ 900

3x+y$\ge$ 9....(3)

Let the minimum cost be Z

so Z=20x+12.50y....(4)

The intersection points of the above lines are (1,6),(4,2),(0,9),(8,0)

Value of Z at(1,6)=

$=20\times1+12.5\times 6$

=95

similarly at (4,2) Z=105

at (0,9) Z=112.5

at(8,0) Z=160

Hence the value of Z is

minimum at (1,6)

So we have to produce 1 kg of Whey protein and 6 kg of Vegan protein to minimize the cost.

The minimize cost is $95