Question

solver: A dairy company gets milk from two dairies and then blends the milk to get the desired amount of butterfat. Milk from dairy I costs $2.40 per gallon, and milk from dairy II costs $0.80 per gallon. At most $144 is available for purchasing milk. Dairy I can supply at most 50 gallons averaging 3.7% butterfat, and dairy II can supply at most 90 gallons averaging 2.9% butterfat. Answer parts a and b. a. How much milk from each supplier should the company buy to get at most 100 gallons of milk with the maximum amount of butterfat? The company should buy nothing gallons from dairy I and nothing gallons from dairy II.

Answer 1

The company should buy 40 gallons from dairy I and 60 gallons from dairy II.

Step-by-step explanation:

Let x represent the number of gallons of dairy I milk and y represent the number.

Since the company can buy at most 100 gallons of milk, hence:

x + y ≤ 100 (1)

The company can spend at most $144, hence:

2.4x + 0.8y ≤ 144 (2)

Dairy I can supply at most 50 gallons and dairy II can supply at most 90 gallons. Hence:

0 ≤ x ≤ 50, 0 ≤ y ≤ 90

The graph was plotted using geogebra. The solution to the problem is at:

(10, 90), (40, 60), (50, 30).

The amount of butterfat is: 0.037x + 0.029y, we are to look for the point with the maximum butterfat.

At (10, 90): total butterfat = 0.037(10) + 0.029(90) = 2.98

At (40, 60): total butterfat = 0.037(40) + 0.029(60) = 3.22

At (50, 30): total butterfat = 0.037(50) + 0.029(30) = 2.72

The company should buy 40 gallons from dairy I and 60 gallons from dairy II.