B Apply

14. Think About a Plan A biologist is developing two new strains of bacteria. Each sample of Type I bacteria produces four new viable bacteria, and each sample of Type II produces three new viable bacteria. Altogether, at least 240 new viable bacteria must be produced. At least 30, but not more than 60, of the original samples must be Type I. Not more than 70 of the original samples can be Type II. A sample of Type I costs $5 and a sample of Type II costs $7. How many samples of Type II bacteria should the biologist use to minimize the cost?
    • What are the unknowns?
    • What constraints do you get from each condition in the problem?
    • Are there any implicit constraints?

15. Error Analysis Your friend is trying to find the maximum value of P = − x + 3 y p equals negative x plus 3 y  subject to the following constraints.

    { y ≤ − 2 x + 6 y ≤ x + 3 x ≥ 0 , y ≥ 0 left brace . table with 3 rows and 1 column , row1 column 1 , y less than or equal to negative 2 x plus 6 , row2 column 1 , y less than or equal to x plus 3 , row3 column 1 , x greater than or equal to 0 comma y greater than or equal to 0 , end table

    

    What error did your friend make? What is the correct solution?

16. Cooking Baking a tray of corn muffins takes 4 cups of milk and 3 cups of wheat flour. Baking a tray of bran muffins takes 2 cups of milk and 3 cups of wheat flour. A baker has 16 cups of milk and 15 cups of wheat flour. He makes $3 profit per tray of corn muffins and $2 profit per tray of bran muffins. How many trays of each type of muffin should the baker make to maximize his profit?

Graph each system of constraints. Name all vertices. Then find the values of x and y that maximize or minimize the objective function. Find the maximum or minimum value.

17. { 3 x + y ≤ 7 x + 2 y ≤ 9 x ≥ 0 , y ≥ 0 left brace . table with 3 rows and 1 column , row1 column 1 , 3 x plus y less than or equal to 7 , row2 column 1 , x plus 2 y less than or equal to 9 , row3 column 1 , x greater than or equal to 0 comma y greater than or equal to 0 , end table

    Maximum for P = 2x + y

18. { 25 ≤ x ≤ 75 y ≤ 110 8 x + 6 y ≥ 720 left brace . table with 3 rows and 1 column , row1 column 1 , 25 less than or equal to x less than or equal to 75 , row2 column 1 , y less than or equal to 110 , row3 column 1 , 8 x plus 6 y greater than or equal to 720 , end table

    Minimum for C = 8x + 5y

19. { x + y ≤ 11 2 y ≥ x x ≥ 0 , y ≥ 0 left brace . table with 3 rows and 1 column , row1 column 1 , x plus y less than or equal to 11 , row2 column 1 , 2 y greater than or equal to x , row3 column 1 , x greater than or equal to 0 comma y greater than or equal to 0 , end table

    Maximum for P = 3x + 2y

20. { 2 x + y ≤ 300 x + y ≤ 200 x ≥ 0 , y ≥ 0 left brace . table with 3 rows and 1 column , row1 column 1 , 2 x plus y less than or equal to 300 , row2 column 1 , x plus y less than or equal to 200 , row3 column 1 , x greater than or equal to 0 comma y greater than or equal to 0 , end table

    Maximum for P = x + 2 y

21. { 5 x + y ≥ 10 x + y ≤ 6 x + 4 y ≥ 12 x ≥ 0 , y ≥ 0 left brace . table with 4 rows and 1 column , row1 column 1 , 5 x plus y greater than or equal to 10 , row2 column 1 , x plus y less than or equal to 6 , row3 column 1 , x plus 4 y greater than or equal to 12 , row4 column 1 , x greater than or equal to 0 comma y greater than or equal to 0 , end table

    Minimum for C = 10,000 x + 20,000y

22. { 6 ≤ x + y ≤ 13 x ≥ 3 y ≥ 1 left brace . table with 3 rows and 1 column , row1 column 1 , 6 less than or equal to x plus y less than or equal to 13 , row2 column 1 , x greater than or equal to 3 , row3 column 1 , y greater than or equal to 1 , end table

    Maximum for P = 4x + 3y

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