Assume that each gallon of Deluxe Vanilla uses 2 quarts of milk, 2 quarts of cre
ID: 3146601 • Letter: A
Question
Assume that each gallon of Deluxe Vanilla uses 2 quarts of milk, 2 quarts of cream, and 3 ounces of vanilla; each gallon of Regular Vanilla uses 3 quarts of milk, 1 quart(s) of cream, and 2 ounces of vanilla; and each gallon of Deluxe Chocolate uses 3.5 quarts of milk, 0.5 quarts of cream, and 6 ounces of cacao. Assume also that the shop has on hand 194 gallons of milk, 94 gallons of cream, 31 pounds of vanilla, and 42 pounds of cacao.
How many gallons of each type of ice cream should the shop make to use up all of these ingredients? 3
When you setup a system of linear equations to solve this problem, how many variables and how many equations should you have.? 4
Let
x be the amount of Deluxe Vanilla to be produced measured in gallons,
y be the amount of Regular Vanilla to be produced measured in gallons, and
z be the amount of Deluxe Chocolate to be produced measured in gallons.
Formulate the system of linear equations that you must solve:
milk used (in quarts) = 2x+3y+3.5z = ?
cream used (in quarts) = 2x+1y+.5z = ?
vanilla used (in ounces) = 3x+ 2y + 0 = ?
cacao used (in ounces) = 0x+0y+6z = ?
Explanation / Answer
First we must know the following conversions
1 gal=4qt. ;1qt. = 32oz ; and 1lb= 16oz.
Since the answer needs to be gallons, then you can use the modified relationships when doing the conversions :
1gal = 4 qt = 128 oz =8 lb.
Assign letters to the contents
a=milk , b= cream , c= vanilla , d = cacao
The equation in gal look like:
×= (2/4)a + (2/4)b + (3/128)c
y= (3/4)a + (1/4)b + (2/128)c
z= (3.5/4)a + (0.5/4)b + (6/128)d
Collect the terms from each equation above like so. Don't forget about conversion of lb to gal:
194= [(2/4) + (3/4) + (3.5/4)]a => 142= (8.5/4)a => a=66.82
94=[(2/4) + (1/4) + (0.5/4)]b => 94= (3.5/4)b => b=107.43
(31/8)= [(3/128) + (2/128)]c => (31/8)= (5/128)c => c=99.2
(42/8) = (6/128)d => (42/8)*(128/6) = d= 112
These values (a,b,c,d) are now to be plugged into the equations x, y and z to get
x= 89.45 = 89.5 gal of Deluxe Vanilla
y=78.5225= 78.5 gal of Regular Vanilla
z=77.146= 77.1 gal of Deluxe Chocolate