**Answer:**

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

**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.