I am having a lot of problems with this linear programming problem. CAn someone help?

A farmer buys two varieties of animal feed. Type A contains 9 oz. of oats and 2 oz. of corn per pound. Type B contains 5 oz of oats and 10 oz of corn per pound. The farmer wants to combine the two feeds so that the resulting mixture has at least 90 lbs. of oats and at least 60 lbs. of corn. The farmer wants to buy not more than 310 lbs. of Type A and not more than 198 lbs. of Type B feed. Type A fee costs him .18/lb. and Type B feed costs him .19/lb. How many pounds of each type should the farmer buy to minimize the cost?

Here is what I have so far:

My minimum cost equation: .18A + .19B

Here are the constraints so far:

A is greater than or equal to 0
B is greater than or equal to 0
A is less than or equal to 310
B is less than or equal to 198

I am having trouble with the other constraints. How do I fit the number of oz. of corn and oats and the 90 lbs. oats and 60 lbs. corn?

Oh, linear programming problems, they can be real cornfusing! But don't worry, I'm here to add a little humor and help you out!

To solve this problem, let's break it down piece by piece. We have two main constraints - the number of ounces of oats and corn, and the minimum amounts of oats and corn required.

For the ounces, we'll calculate that based on the pounds of feed:
Oats in Type A = 9 oz/lb * A
Corn in Type A = 2 oz/lb * A
Oats in Type B = 5 oz/lb * B
Corn in Type B = 10 oz/lb * B

Now let's tackle the minimum amounts. We want at least 90 pounds of oats and 60 pounds of corn, which translates to:
Oats in Type A + Oats in Type B >= 90
Corn in Type A + Corn in Type B >= 60

Based on these constraints, we can rewrite our updated set of constraints:

Oats constraint: 9A + 5B >= 90
Corn constraint: 2A + 10B >= 60

Now, let's put it all together. We have our cost equation (.18A + .19B) and our constraints:

A is greater than or equal to 0
B is greater than or equal to 0
A is less than or equal to 310
B is less than or equal to 198
Oats constraint: 9A + 5B >= 90
Corn constraint: 2A + 10B >= 60

With this information, you can plug it into your linear programming solver and minimize the cost. Remember, there's always a solution to every problem, just like there's always a kernel of truth in every corny joke!

To establish the constraints based on the number of ounces of oats and corn, as well as the minimum requirements for oats and corn, you can set up the following equations:

1. Oats constraint:
9A + 5B ≥ 90
This equation represents the total ounces of oats in Type A and Type B feed, which should be equal to or greater than 90 lbs.

2. Corn constraint:
2A + 10B ≥ 60
This equation represents the total ounces of corn in Type A and Type B feed, which should be equal to or greater than 60 lbs.

Now, let's add these constraints to your existing list:

3. A ≥ 0
4. B ≥ 0
5. A ≤ 310
6. B ≤ 198
7. 9A + 5B ≥ 90
8. 2A + 10B ≥ 60

To summarize, the constraints for your linear programming problem are:

A ≥ 0
B ≥ 0
A ≤ 310
B ≤ 198
9A + 5B ≥ 90
2A + 10B ≥ 60

You can now use these constraints along with your objective function (.18A + .19B) to solve the linear programming problem and determine the quantities of Type A and Type B feed the farmer should buy to minimize the cost.

To determine the remaining constraints, you need to consider the number of ounces of oats and corn in each pound of feed and the desired minimum quantities of oats and corn in the resulting mixture.

Let's start with the oats constraint. We know that Type A feed contains 9 oz. of oats per pound, while Type B feed contains 5 oz. of oats per pound. Let's assume we use A pounds of Type A feed and B pounds of Type B feed.

The total oats in the resulting mixture would then be 9A + 5B ounces. To convert this to pounds, we divide by 16 (since there are 16 ounces in a pound). So, the constraint for oats can be written as:

(9A + 5B)/16 ≥ 90

Next, let's consider the corn constraint. Type A feed contains 2 oz. of corn per pound, while Type B feed contains 10 oz. of corn per pound. Using the same assumption as before, we get:

(2A + 10B)/16 ≥ 60

Combining all the constraints so far, we have:

A ≥ 0
B ≥ 0
A ≤ 310
B ≤ 198
(9A + 5B)/16 ≥ 90
(2A + 10B)/16 ≥ 60

These constraints ensure that the resulting feed mixture contains at least 90 lbs. of oats and 60 lbs. of corn. Now, you can include these constraints in your linear programming problem and solve for the optimal solution using the cost equation you provided: .18A + .19B.