Math, microeconomics ,,, help please

Question

Nutrition   Ruff, Inc. makes dog food out of chicken and grain. Chicken has 10 grams of protein and 5 grams of fat per ounce, and grain has 2 grams of protein and 2 grams of fat per ounce. A bag of dog food must contain at least 226 grams of protein and at least 166 grams of fat. If chicken costs 9¢ per ounce and grain costs 1¢ per ounce, how many ounces of each should Ruff use in each bag of dog food to minimize cost? (If an answer does not exist, enter DNE.)

grain ...... oz

chicken ...... oz

Answer 1

10P +5F = C = 1 chicken ounce

2P +2F = G = 1 grain ounce

chicken costs 9 cents per ounce

grain costs 1 cent per ounce

how to minimize cost?

bag of dog food requires minimum 226P + 166G

10C +2G = 226

5C + 2G = 166

subtract

5C = 60

C =60/5= 12 ounces of Chicken

G= (166-60)/2 =106/2 = 53 ounces of Grain

costing 9(12)+53 = 108+53 = $1.62 per bag of Ruff's dog food

which provides exactly the minimum required grams of P and F

Grain is 9 times cheaper, so another alternative is use as much grain as possible to satisfy both P and F

266/2 = 133 ounces of Grain for 266 P

166/2 = 83 ounces of Grain for 166 F

133 ounces of Grain cost $1.33 and has enough P and more than enough F at lower cost

or try all Chicken

266/10 = 26.6 ounces of Chicken for 266 P

166/5 = 33.2 ounces of Chicken for 166 F

33.2 ounce of Chicken cost 9(33.2) = about $2.99

Minimize costs by using 133 ounces of Grain and zero Chicken

the more Grain used, and the less Chicken, the cheaper the dog food costs,

although the dog may turn up its nose at all grain