Thread: I need help finding constraints for this word problem for math

the dogs in kens k-9 kennel must get at least 40 pounds of food per day. the food may be a mixture of foods x and y. food x has 1 unt per pound of nutrient a and 1/2 unit per pound of nutrient b. food y has 1/3 unit per pound of nutrient a and 1 unit per pound of nutrient b. the daily diet must include at peast 20 units of nutrient A and at least 30 units of nutrient b. the dogs must not get more than 100 pounds of food per day. food x costs $.80 per pound and food y costs $.40 per pound. what is the least possible cost per day for feeding the dogs?

Fix units of: pounds for weight, dollars for money, and "units of nutrients" for nutrients.

Let u = number pounds of food x, and v = number of pounds of food y.
Let a = units of nutrient A received, and b = units of nutrient B received.
Let C = daily cost in dollars of feeding a dog.

"get at least 40 pounds of food per day" becomes:
u+v>=40.

"food x has 1 unt per pound of nutrient a and 1/2 unit per pound of nutrient b. y has 1/3 unit per pound of nutrient a and 1 unit per pound of nutrient b" becomes:
a = (1)u + (1/3)v
b = (1/2)u + (1)v

"the daily diet must include at peast 20 units of nutrient A and at least 30 units of nutrient b" becomes:
a >= 20
b >= 30

"the dogs must not get more than 100 pounds of food per day." becomes:
u+v <= 100

"food x costs $.80 per pound and food y costs $.40 per pound" becomes:
C = (4/5)u + (2/5)v

"what is the least possible cost per day for feeding the dogs?" becomes:
Minimize C subject to all those constraints.

Implicit constraints:
u>=0, v>=0

Putting it all together:
u>=0, v>=0
u+v>=40.
a = (1)u + (1/3)v
b = (1/2)u + (1)v
a >= 20
b >= 30
u+v <= 100
C = (4/5)u + (2/5)v
Minimize C subject to all those constraints.

Problem Translated:
(1)u + (1/3)v >=20
(1/2)u + (1)v >=30
u>=0, v>=0
40<= u+v <= 100
C = (4/5)u + (2/5)v
Minimize C subject to all those constraints.