Question

A diet meal is to contain at least 7 units of vitamins, 5 units of minerals, and 11 calories.Two sets of foods are to be purchased for the said meal. Set A provides 2 units ofvitamins, 1 unit of minerals, and 1 calorie. Set B provides 1 unit of vitamins, 2 units ofminerals, and 3 units of calories. If set A costs P180 per unit and set B costs P300 perunit, how many units of each set of foods should be purchased to minimize the cost?

Answer 1

Let the quantity of set A be represented by a units and the quantity of set B be represented by b units. From the question we have that the following constraints hold

$a\ge0\\b\ge0\\2a+b\ge7\\a+2b\ge5\\a+3b\ge11$

Since we need to minimize the cost. Hence we have function

$minz = 180a +300b$

Combining all constraints we have that

$minz = 180a +300b$ subject to the following constraints

$a\ge0\\b\ge0\\2a+b\ge7\\a+2b\ge5\\a+3b\ge11$

Using corner method in the graph below

we have that the corner points are

$(0,7), (2,3),(11,0)$

Hence we input each corner point into our cost minimization function

$minz = 180a +300b$

$\text{Since (2,3) has the least cost, therefore we need 2 sets of food A and 3 sets of food B}$