Answer:
a) i The company should buy 40 gallons from dairy I and 60 gallons from dairy
ii) What is the maximum amount of butterfat? The total amount of butterfat from Diary I and Diary II = 3.12% + 1.93%
=5.05%
b.The excess capacity of dairy I is 10 gallons, and for dairy II it is 30 gallons.
Explanation:
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?
From the question, we are told that:
Milk from dairy I costs $2.40 per gallon, Milk from dairy II costs $0.80 per gallon.
Let's represent:
Number of gallons of Milk from dairy I = x
Number of gallons of Milk from dairy II = y
At most $144 is available for purchasing milk.
$2.40 × x + $0.80 × y = 144
2.40x + 0.80y = 144........ Equation 1
x + y = 100....... Equation 2
x = 100 - y
2.40(100 - y) + 0.80y = 144
240 - 2.4y + 0.80y = 144
-1.60y = 144 - 240
-1.6y = -96
y = -96/-1.6
y = 60
From Equation 2
x + y = 100....... Equation 2
x + 60 = 100
x = 100 - 60
x = 40
Therefore, since number of gallons of Milk from dairy I = x and number of gallons of Milk from dairy II = y
The company should buy 40 gallons from dairy I and 60 gallons from dairy
II. What is the maximum amount of butterfat?
From the question
Dairy I can supply at most 50 gallons averaging 3.9% butterfat,
50 gallons = 3.9% butterfat
40 gallons =
Cross Multiply
= 40 × 3.9/50
= 3.12%
Dairy II can supply at most 90 gallons averaging 2.9% butterfat.
90 gallons of milk = 2.9% butter fat
60 gallons of milk =
Cross Multiply
= 60 × 2.9%/90
=1.9333333333%
≈ 1.93%
The total amount of butterfat from Diary I and Diary II = 3.12% + 1.93%
=5.05%
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.
Excess capacity of Diary I =
50 gallons - 40 gallons = 10 gallons
Excess capacity of Diary II =
90 gallons - 60 gallons = 30 gallons
Therefore, the excess capacity of dairy I is 10 gallons, and for dairy II it is 30 gallons.