Sheila knows that the minimum daily requirements for her dog's diet are 18 units of protein and 30 units of carbohydrates. Pet dog food costs $1.65 a can, contains 2 units of protein and 6 units of carbohydrates. Budget dog dood costs $1.35 a can, contains 2 units of protein and 2 units of carbohydrates. What combination of the two brands of dog food should sheila give her dog each day to ensure it has a proper diet at minimum cost?

This problem involves finding the optimal number of cans of each type of dog food to minimize Sheila's cost while satisfying the protein and carbohydrate requirements. We can use linear programming to solve this.

### Let:
- $ x_1 $ = number of cans of Pet dog food.
- $ x_2 $ = number of cans of Budget dog food.

### Objective Function:
Sheila wants to minimize the total cost. The cost is:
\[
\text{Cost} = 1.65x_1 + 1.35x_2
\]

### Constraints:
The dog's diet needs to meet the following daily minimum requirements:

1. **Protein constraint**: The dog needs at least 18 units of protein. Both foods provide 2 units of protein per can:
\[
2x_1 + 2x_2 \geq 18
\]
Simplifying this:
\[
x_1 + x_2 \geq 9
\]

2. **Carbohydrate constraint**: The dog needs at least 30 units of carbohydrates. Pet dog food provides 6 units per can, and Budget dog food provides 2 units per can:
\[
6x_1 + 2x_2 \geq 30
\]

3. **Non-negativity constraint**: Since the number of cans can't be negative:
\[
x_1 \geq 0, \quad x_2 \geq 0
\]

### Solving the System:
We will solve this using a combination of the constraints.

1. From the protein constraint:
   \[
   x_1 + x_2 \geq 9
   \]
   So, $ x_2 = 9 - x_1 $, for now.

2. Substitute into the carbohydrate constraint:
   \[
   6x_1 + 2(9 - x_1) \geq 30
   \]
   Expanding:
   \[
   6x_1 + 18 - 2x_1 \geq 30
   \]
   Simplifying:
   \[
   4x_1 \geq 12
   \]
   \[
   x_1 \geq 3
   \]

3. If $ x_1 = 3 $, then $ x_2 = 9 - 3 = 6 $.

### Conclusion:
Sheila should buy 3 cans of Pet dog food and 6 cans of Budget dog food each day to meet the nutritional requirements at the minimum cost.

- **Cost**: 
  \[
  1.65(3) + 1.35(6) = 4.95 + 8.10 = 13.05
  \]

The minimum cost is $13.05 per day.

Would you like a more detailed breakdown or to explore other scenarios?

Here are 5 questions you could explore next:
1. How would the result change if the prices of the dog foods were different?
2. What if the dog’s protein or carbohydrate requirements increased?
3. Could you use another optimization method, like graphing the inequalities?
4. How would you solve this problem if more types of dog food were available?
5. How much more would it cost if Sheila only used Pet dog food?

**Tip**: When solving linear programming problems, always check if the constraints are feasible by trying different values within the constraints.

Sheila knows that the minimum daily requirements for her dog's diet are 18 units of protein and 30 units of carbohydrates. Pet dog food costs $1.65 a can, contains 2 units of protein and 6 units of carbohydrates. Budget dog food costs $1.35 a can, contains 2 units of protein and 2 units of carbohydrates. What combination of the two brands of dog food should Sheila give her dog each day to ensure it has a proper diet at minimum cost?

Learn how to solve a linear programming problem to minimize the cost of dog food while meeting nutritional requirements. Sheila's dog needs 18 units of protein and 30 units of carbohydrates daily. Find the optimal combination of Pet dog food and Budget dog food to achieve this at the minimum cost using algebra and optimization methods.

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