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 224 grams of protein and at least 174 grams of fat. If chicken costs 11 per ounce and grain costs 2 per ounce, how many ounces of each should Ruff use in each bag of dog food in order to minimize cost?

whoops, need to be above and to the right of the lines so corners are

(0,112) (34.8, 0) (10,62)
first 2*112= 224
second 234 again
third 11*34.8 = 382.8
so I get all grain

To minimize cost, we need to determine the number of ounces of chicken and grain that will meet the protein and fat requirements while minimizing the cost.

Let's assume x represents the number of ounces of chicken and y represents the number of ounces of grain.

We have the following constraints:
- Protein constraint: 10x + 2y ≥ 224
- Fat constraint: 5x + 2y ≥ 174

The objective is to minimize the cost, which is given by the following equation:
Cost = 11x + 2y

To solve this problem, we can use linear programming.

First, let's rewrite the constraints in terms of a single variable:
- Protein constraint: x ≥ (224 - 2y) / 10
- Fat constraint: x ≥ (174 - 2y) / 5

Now, we can plot the feasible region on a graph and find the corner points.

To find the corner points, we need to solve for x in the protein and fat constraints:

For the protein constraint: x ≥ (224 - 2y) / 10
Multiply both sides by 10:
10x ≥ 224 - 2y
10x + 2y ≥ 224

For the fat constraint: x ≥ (174 - 2y) / 5
Multiply both sides by 5:
5x ≥ 174 - 2y
5x + 2y ≥ 174

Now, let's plot these constraints on the graph:

- The protein constraint, 10x + 2y ≥ 224, can be simplified to y ≥ (224 - 10x) / 2.
- The fat constraint, 5x + 2y ≥ 174, can be simplified to y ≥ (174 - 5x) / 2.

Looking at the graph, we can see that the corner points are (0,87), (35,39), (42,31), and (56,0).

Next, substitute the values of x and y into the cost equation to find the cost at each corner point:

- Cost at (0,87): Cost = 11(0) + 2(87) = 174
- Cost at (35,39): Cost = 11(35) + 2(39) = 609
- Cost at (42,31): Cost = 11(42) + 2(31) = 604
- Cost at (56,0): Cost = 11(56) + 2(0) = 616

Therefore, the minimum cost is 604, which occurs when 42 ounces of chicken and 31 ounces of grain are used in each bag of dog food.

To solve this problem, we need to find the optimal combination of chicken and grain that minimizes cost while meeting the protein and fat requirements.

Let's assume that Ruff uses x ounces of chicken and y ounces of grain in each bag of dog food.

The cost of chicken per ounce is $11, so the total cost of chicken used is 11x dollars.

The cost of grain per ounce is $2, so the total cost of grain used is 2y dollars.

We need to minimize the cost, which is represented by the function C(x, y) = 11x + 2y.

Now let's look at the nutritional requirements. Chicken has 10 grams of protein per ounce, so the total protein from chicken used is 10x grams. Grain has 2 grams of protein per ounce, so the total protein from grain used is 2y grams.

Similarly, chicken has 5 grams of fat per ounce, so the total fat from chicken used is 5x grams. Grain has 2 grams of fat per ounce, so the total fat from grain used is 2y grams.

We need to meet the protein and fat requirements, which can be written as the following two inequalities:

10x + 2y ≥ 224 (protein requirement)
5x + 2y ≥ 174 (fat requirement)

Now we have a system of inequalities:
10x + 2y ≥ 224
5x + 2y ≥ 174

To solve this system, we can use the graphical method or the simplex algorithm. However, since this is a linear programming problem, we can solve it using linear programming techniques.

The objective is to minimize C(x, y) = 11x + 2y, subject to the constraints:
10x + 2y ≥ 224
5x + 2y ≥ 174

Using linear programming software or techniques, the optimal values for x and y are obtained. Once we have these values, we can determine the optimal combination of chicken and grain for the dog food.

Therefore, to find the number of ounces of each that Ruff should use, it is recommended to use linear programming techniques or consult a professional who specializes in linear programming.

a) 10 c + 2 g >/= 224

b) 5 c + 2 g >/= 174

c) minimize p = 11 c + 2 g

graphs c on x axis, g on y
a) (0,112) (22.4,0)
b) (0,87) (34.8,0)
intersection of lines a and b
10 c + 2 g = 224
10 c + 4 g = 348
------------------
-2 g = -124
g = 62
then c = 10
so (10,62)

cost line p = 11 c + 2 g
check corners (0,87)(10,62)(12.4,0)
first p = 2*87 = 174
second p = 110+124 = 234
third p = 11*12.4 = 136.4
so I get all chicken