14.-7.18 points TanFin12 3.3.054. My Notes A veterinarian has been asked to prep

Question

14.-7.18 points TanFin12 3.3.054. My Notes A veterinarian has been asked to prepare a diet, x ounces of Brand A and y ounces of Brand B, for a group of dogs to be used in a nutrition study at the School of Animal Science. It has been stipulated that each serving should be no larger than 8 oz and must contain at least 29 units of Nutrient I and 20 units of Nutrient II. The vet has decided that the diet may be prepared from two brands of dog food: Brand A and Brand B. Each ounce of Brand A contains 3 units of Nutrient I and 4 units of Nutrient II. Each ounce of Brand B contains 5 units of Nutrient I and 2 units of Nutrient II. Brand A costs 5 cents/ounce, and Brand B costs 6 cents/ounce. Determine how many ounces of each brand of dog food should be used per serving to meet the given requirements at a minimum cost. (x, y) What is the minimum cost? (Round your answer to the nearest cent.) cents per serving Need Help? Read It

Explanation / Answer

To do this, we need to set up and solve the LP.First,

Let x=ounces of Brand A used

y=ounces of Brand B used

Next we need our objective function and constraints:

Minimize cost, C= 5x + 6y

subject to

x + y <= 8

3x + 5y >= 29

4x + 2y >= 20

x , y >= 0

With linear programming, we know that the vertices of the feasible space are the only possible solutions for the objective function. So the next thing we have to do is determine our feasible space. We can do this by graphing the constraints. To start x,y>=0 has the x - and y-axis as boundaries and means the answer has to be in first quadrant. Next, x+y <=8 is the line from (0,8) to (8,0) and is the area underneath the graph towards the origin since the inequality is less than.

The line 3x + 5y>= 29 goes through (0, 5.8) , (29/3 , 0) and is the area above the line since the inequality is greater than. 4x+2y>= 20 goes through (0,10) and (5,0) and is the area above the line since the inequality is greater than.

We now have our feasible space. It is bounded by (2,6) , (2.5,5.5) and (3,4)

Now that we have all the points, we plug them into the objective function:

C=5(2) +6(6)=46

C=5(3) +6(4)=39

C=5(5.5)+ 6(2.5)=42.5

We see that our costs are minimized when we have 3 ounces of brand A and 4 ounces of brand B for 39 cents/saving

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