SOLUTION: Hugo is making a protein shake. He will combine two different shake powders: Powder 1 and Powder 2. One scoop of Powder 1 contains 3 grams of carbohydrates, 22 grams of fat, and 2

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: Hugo is making a protein shake. He will combine two different shake powders: Powder 1 and Powder 2. One scoop of Powder 1 contains 3 grams of carbohydrates, 22 grams of fat, and 2      Log On


   

Question 1135146: Hugo is making a protein shake. He will combine two different shake powders: Powder 1 and Powder 2. One scoop of Powder 1 contains 3 grams of carbohydrates, 22 grams of fat, and 240 calories. One scoop of Powder 2 contains 6 grams of carbohydrates, 10 grams of fat, and 80 calories. If Hugo wants his shake to have at least 3600 calories and at most 90 grams of carbohydrates, how many scoops of each powder should he use to make a shake with as little fat as possible?

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Let x and y be the number of scoops of powders 1 and 2, respectively.

The constraints in the problem are

(1) at least 3600 calories:
240x%2B80y+%3E=+3600

(2) at most 90g of carbs:
3x%2B6y+%3C=+90

And of course x and y are both non-negative.

A graph of the constraint boundary lines: red = calories; green = carbs

graph%28400%2C400%2C-9%2C40%2C-8%2C50%2C45-3x%2C15-.5x%29

The intersection of the constraint boundary lines is (12,9).

We need AT LEAST 3600 calories, so we need to be ABOVE the red line; and we need AT MOST 90g of carbs, so we need to be BELOW the green line.

That makes the coordinates of the vertices of the feasibility region (15,0), (30,0), and (12,9).

The objective is to minimize the amount of fat; the amount of fat (the objective function) is 22x+10y.

Evaluate that expression at each vertex of the feasibility region to find the number of scoops of each powder that will satisfy all the requirements.