A company makes two types of​ biscuits: Jumbo and Regular. The oven can cook at most 400 biscuits per day. Each jumbo biscuit requires 2 oz of​ flour, each regular biscuit requires 1 oz of​ flour, and there is 600 oz of flour available. The income from each jumbo biscuit is ​$0.06 and from each regular biscuit is ​$0.11. How many of each size biscuit should be made to maximize​ income? What is the maximum​ income?
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Part 1
The company should make 
  
enter your response here jumbo and 
  
enter your response here regular biscuits.
​(Type whole​ numbers.)
Part 2
The maximum income is ​$
  
enter your response here.
​(Type an integer or decimal rounded to two decimal places as​ needed.)

To solve this problem, we need to maximize the company's income by determining the optimal number of jumbo and regular biscuits to produce, given the constraints of oven capacity and flour availability.

### Step 1: Define the variables
Let:
- $ x $ = number of jumbo biscuits made
- $ y $ = number of regular biscuits made

### Step 2: Write the objective function
The objective is to maximize the income:
\[
\text{Income} = 0.06x + 0.11y
\]

### Step 3: Write the constraints
1. **Oven capacity constraint:** The total number of biscuits cannot exceed 400:
\[
x + y \leq 400
\]

2. **Flour constraint:** The total amount of flour used cannot exceed 600 oz:
\[
2x + y \leq 600
\]

3. **Non-negativity constraint:** The number of biscuits cannot be negative:
\[
x \geq 0, \quad y \geq 0
\]

### Step 4: Graph the constraints and find the feasible region
We will graph the inequalities to find the feasible region, which will be a polygon in the first quadrant. The vertices of this polygon will give us the possible solutions.

### Step 5: Determine the vertices of the feasible region
The vertices occur where the constraints intersect. We'll solve the system of equations:

1. **Intersection of** $ x + y = 400 $ **and** $ 2x + y = 600 $:
   \[
   x + y = 400 \quad \text{(Equation 1)}
   \]
   \[
   2x + y = 600 \quad \text{(Equation 2)}
   \]
   Subtract Equation 1 from Equation 2:
   \[
   2x + y - (x + y) = 600 - 400
   \]
   \[
   x = 200
   \]
   Substitute $ x = 200 $ back into Equation 1:
   \[
   200 + y = 400 \quad \Rightarrow \quad y = 200
   \]
   So, one vertex is $ (200, 200) $.

2. **Intersection of** $ x + y = 400 $ **and** $ x = 0 $:
   \[
   0 + y = 400 \quad \Rightarrow \quad y = 400
   \]
   So, another vertex is $ (0, 400) $.

3. **Intersection of** $ 2x + y = 600 $ **and** $ y = 0 $:
   \[
   2x + 0 = 600 \quad \Rightarrow \quad x = 300
   \]
   So, another vertex is $ (300, 0) $.

### Step 6: Evaluate the objective function at each vertex
Substitute the vertices into the income function $ 0.06x + 0.11y $:

1. At $ (200, 200) $:
   \[
   \text{Income} = 0.06(200) + 0.11(200) = 12 + 22 = 34
   \]

2. At $ (0, 400) $:
   \[
   \text{Income} = 0.06(0) + 0.11(400) = 44
   \]

3. At $ (300, 0) $:
   \[
   \text{Income} = 0.06(300) + 0.11(0) = 18
   \]

### Step 7: Identify the maximum income
The maximum income is $ \$44 $, which occurs when $ x = 0 $ and $ y = 400 $.

### Final Answer:
- The company should make **0 jumbo biscuits** and **400 regular biscuits**.
- The maximum income is **$44.00**.

Would you like to see more details on the steps or have any questions? Here are some related questions you might find useful:

1. How would the solution change if the income from a jumbo biscuit was $0.15 instead of $0.06?
2. What happens if the flour available is reduced to 500 oz?
3. How would the constraints change if the oven capacity was increased to 500 biscuits per day?
4. Can this problem be solved using the Simplex method instead of graphing?
5. How does changing the number of jumbo biscuits impact the income if the flour constraint is removed?

**Tip:** In linear programming problems, the solution typically occurs at one of the vertices of the feasible region, so checking those points systematically can help identify the optimal solution efficiently.

A company makes two types of biscuits: Jumbo and Regular. The oven can cook at most 400 biscuits per day. Each jumbo biscuit requires 2 oz of flour, each regular biscuit requires 1 oz of flour, and there is 600 oz of flour available. The income from each jumbo biscuit is $0.06 and from each regular biscuit is $0.11. How many of each size biscuit should be made to maximize income? What is the maximum income?

Explore how to maximize income from producing Jumbo and Regular biscuits using linear programming. Learn to set up and solve optimization problems with constraints involving oven capacity and flour availability. This problem is ideal for high school students studying algebra and optimization.

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