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 $2.40per gallon, and milk from dairy II costs $0.80per gallon. At most $144 is available for purchasing milk. Dairy I can supply at most 50 gallons averaging 3.9 % 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 amount of butterfat?

The company should buy____gallons from dairy I and____gallons from dairy II.

What is the maximum amount of butterfat? (Type an integer or a decimal.)

b. The solution from part a leaves both dairy I and dairy II with excess capacity. Calculate the amount of additional milk each dairy could produce.

The excess capacity of dairy I is______gallons, and for dairy II it is____gallons.

c. Is there any way all this capacity could be used while still meeting the other constraints? Explain.

A. Yes,10 more gallons can be bought from dairy I without going over budget.

B. Yes,10more gallons can be bought from dairy I and 30 more from dairy II without going over budget.

C. Yes,10 more gallons can be bought from dairy I without going over budget.

D. No. Any more milk purchased from either dairy will go over budget.

Answer 1

consider the drain organization gets milk from two dairies

give x a chance to be the quantity of gallons from dairy I and y be the quantity of gallons from dairy II

build the obliges as pursues

milk from dairy I cost 2.40 Dollar per gallon , and drain from dairy II costs 0.80 Dollar per gallon. at most 144 is accessible for unadulterated milk.

that is 2.40x+0.80y≤144

i.e 0.60x+0.20y≤36

dairy I can supply at most 50 gal of milk i.e x ≤ 50

dairy II can supply at most 90 gal of milk i.e y ≤ 90

the organization shouldd get at most 100 gal of milk from every provider i.e x+y≤100

the goal is to locate the most extreme percent of butterfat.

at that point the target work is to boost z=0.039x+0.029y

the linear  programming issue is as per the following:

expand z=0.039x+0.029y

subject to:

0.60x+0.20y ≤ 36

x ≤ 50

y ≤ 90

x+y ≤ 100

x≥0

y≥0

when draw the feasible region and identify the corner points we will get as follows

max{0.039 x + 0.029 y|x + y<=100 && 0.6 x + 0.2 y<=36 && x<=50 && x>=0 && y>=0} = 3.3 at (x, y) = (40, 60)

the most extreme estimation of target work is 3.3 which are gotten at the point (40,60) accordingly , the greatest measure of bufferfat is 3.3% which happens 40 gal of milk from dairy I and 60 gal of milk from dairy II are bought.

b)

in the arrangement from part(a), the organization utilizes 40 gallons from dairy I and dairy I have an ability to deliver 50 galloons of milk.

hence, there is an abundance limit of 10 gallons from dairy I

the organization utilizes 60 gallons from dairy II and dairy II have an ability to create 90 galloons of milk.

along these lines, there is an overabundance limit of 30 gallons from dairy II