Question

Jessica needs at least 60 units of vitamin A, 40 units of vitamin B, and 140 units of vitamin C each week. She can choose between Costless brand or Savemore brand vitamins. A Costless vitamin costs 5 cents and contains 3 units of vitamin A, 1 unit of vitamin B, and 2 units of vitamin C. A Savemore vitamin costs 7 cents and contains 1 unit of A, 1 of B, and 5 of C. Determine the minimum cost for Jessica to purchase all the vitamins she needs.

Answer 1

$2.40

Explanation:

Let x be the number of Costless vitamins and let y be the number of Savemore vitamins. Since we are trying to minimize cost, our objective function represents the cost of buying both Costless and Savemore vitamins. This function is defined by

C(x,y)=0.05x+0.07y

The first constraint is that Jessica needs 60 units of vitamin A and we know that Costless vitamins have 3 units of vitamin A and the Savemore vitamin has 1 unit of Vitamin A. This translates into the following inequality.

3x+y≥60

The second constraint is that Jessica needs at least 40 units of vitamin B and we know that Costless and Savemore vitamins each have 1 unit of vitamin B. This translates into the following inequality.

x+y≥40

The third constraint is that Jessica needs at least 140 units of vitamin C and we know that the Costless vitamin has 2 units of vitamin C and Savemore vitamins has 5 units of vitamin C. This translates into the following inequality.

2x+5y≥140

The fact that x and y must be positive numbers is represented by the following two constraints:

x≥0,y≥0

Using all of this information, our problem is as follows.

Minimize: C(x,y)=0.05x+0.07y

Subject to: 3x+y≥60

x+y≥40

2x+5y≥140

x≥0

y≥0

The corner points are:

(0,60),(70,0),(10,30),(20,20)

To minimize cost, we will substitute these points in the objective function to see which point gives us the lowest number. The results are listed in the table below.

The point (20,20) gives lowest cost: $2.40 .