Word Problem-5

Question

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 55 gallons averaging 3.7% butterfat, and Dairy II can supply at most 90 gallons averaging 2.9% butterfat. 

 

A. How much milk from each supplier should the company buy to get at most 100 gallons of milk with the maximum butterfat?

The company should buy_____gallons from Dairy 1 and ______gallons from Dairy 2

Answer 1

This is a linear programming question with several constraints. Graphing the system and finding intersection points will help out a lot.

 

To start, lets define variables.

Let x be the gallons of milk from dairy I, and

Let y be the gallons of milk from dairy II.

 

Our simplest constraints are from the limits on the amounts that can be purchased:

0 ≤ x ≤ 55

0 ≤ y ≤ 90

0 ≤ x+y ≤ 100

 

The other constraint is cost, since there is a maximum amount to spend. The cost to buy x gallons from dairy I is $2.40x and similarly $0.80y from dairy II.

The total cost is then:

2.40x + 0.80y ≤ 144

 

When you graph all of these constraints, you'll get a 5 or 6 sided region bounded by the x and y axes and several (if not all) of these lines. The maximum will occur at one of the intersections. You can find the points of intersection by solving the appropriate system of two equations (substitution should work best for these.)

 

You're trying to maximize butterfat, so you need an equation to calculate it.

x gallons of milk from dairy I have 0.037x gallons of butterfat, and for dairy II, 0.029y.

The total butterfat is the sum of these 0.037x + 0.029y.

 

Calculate this value for each intersection in your region, and your solution is the one with the highest total.