Question

A public aquarium is adding coral nutrients to a large reef tank. The minimum amounts of nutrients A, B, and C that need to be added to the tank are 30 units, 16 units, and 24 units, respectively. Information about each bottle of brand X and brand Y additives is shown below. How many bottles of each brand must be added to satisfy the needs of the reef tank at the minimum possible cost?

Cost. Nutrient A. Nutrient B. Nutrient C.
Brand X. $25. 3units. 3 units. 7 units
Brand Y. $15. 9 units. 2 units. 2 units​

Answer 1

Answer: 0 bottles of Brand X and 10 bottles of Brand Y

(please let me know if I read the question wrong and I will do my best to fix it)

Explanation:

This problem can be solved using linear programming. Let x be the number of Brand X bottles and y be the number of Brand Y bottles. The objective is to minimize the cost, given by:

Cost = 25x + 15y

Subject to the following constraints, which represent the minimum nutrient requirements:

3x + 9y ≥ 30 (Nutrient A)

3x + 2y ≥ 16 (Nutrient B)

7x + 2y ≥ 24 (Nutrient C)

x ≥ 0, y ≥ 0 (Non-negativity constraint)

We can solve this system of inequalities graphically by finding the feasible region and the corner points.

Nutrient A constraint (3x + 9y ≥ 30):

Divide the inequality by 3:

x + 3y ≥ 10

Nutrient B constraint (3x + 2y ≥ 16):

Divide the inequality by 2:

1.5x + y ≥ 8

y ≥ 8 - 1.5x

Nutrient C constraint (7x + 2y ≥ 24):

Divide the inequality by 2:

3.5x + y ≥ 12

y ≥ 12 - 3.5x

Now we will graph the inequalities on the xy-plane:

y ≥ 10 - x/3

y ≥ 8 - 1.5x

y ≥ 12 - 3.5x

The feasible region is the area where all three inequalities are satisfied. It's a triangular region with corner points A(0,10), B(4,4), and C(8/3, 2).

Now we need to find the cost at each corner point:

Cost A (0, 10) = 25(0) + 15(10) = $150

Cost B (4, 4) = 25(4) + 15(4) = $160

Cost C (8/3, 2) = 25(8/3) + 15(2) ≈ $153.33

The minimum cost is $150, which occurs when 0 bottles of Brand X and 10 bottles of Brand Y are used. So, the public aquarium should add 0 bottles of Brand X and 10 bottles of Brand Y to satisfy the needs of the reef tank at the minimum possible cost.