What are the corners of the region bounded by 5x + 4y ≤ 400 which is not part of the feasible region for the following problem?

You are in the business of baking bread and cake. A loaf of bread is sold for $2 and a cake for $3. Bread takes 5 minutes and cake takes 4 minutes to bake. Bread takes 0.15 pound of flower dough and cake takes 0.6 pound. You have 6 2/3 hours of labor and 43.2 pounds of flower dough.


(0, 0), (0, 100), and (80, 0)

(0, 0), (0, 72), (28, 65), and (80, 0)

(0, 0), (0, 100), (28, 65), and (80, 0)

(0, 72), (0, 100), and (28, 65)

I don't know


To solve this problem, we need to analyze the constraints and identify the feasible region for the problem. Let's break it down step by step.

### Problem Breakdown:

1. **Objective:**
   - Maximize profit from selling bread and cake.
   - Bread sells for $2, cake for $3.

2. **Constraints:**
   - Time: 5 minutes per bread, 4 minutes per cake.
   - Dough: 0.15 pounds per bread, 0.6 pounds per cake.
   - Available time: $6 \frac{2}{3}$ hours = 400 minutes.
   - Available dough: 43.2 pounds.

3. **Inequalities:**
   - Time: $5x + 4y \leq 400$
   - Dough: $0.15x + 0.6y \leq 43.2$
   - Non-negativity: $x \geq 0, y \geq 0$

### Step 1: Analyze the Inequalities

1. **Time Constraint:**
   \[
   5x + 4y \leq 400
   \]

2. **Dough Constraint:**
   \[
   0.15x + 0.6y \leq 43.2
   \]
   This can be simplified by multiplying the entire inequality by 10:
   \[
   1.5x + 6y \leq 432
   \]

### Step 2: Find the Feasible Region

We need to find the region bounded by these inequalities, considering that $x \geq 0$ and $y \geq 0$.

1. **Time Constraint:**
   - Intercept at $x$-axis: $x = \frac{400}{5} = 80$
   - Intercept at $y$-axis: $y = \frac{400}{4} = 100$

2. **Dough Constraint:**
   - Intercept at $x$-axis: $x = \frac{432}{1.5} = 288$
   - Intercept at $y$-axis: $y = \frac{432}{6} = 72$

3. **Feasible Region:**
   The feasible region is bounded by the above constraints. 

### Step 3: Identify the Corners of the Feasible Region

The corners of the feasible region are where the boundary lines intersect.

1. **Intersection of Time Constraint with $x$- and $y$-axes:**
   - (0, 100)
   - (80, 0)

2. **Intersection of Dough Constraint with $x$- and $y$-axes:**
   - (0, 72)
   - (288, 0) (but this point is beyond $x = 80$ so not part of feasible region)
  
3. **Intersection of the Two Constraints:**
   To find the intersection of $5x + 4y = 400$ and $1.5x + 6y = 432$:
   Solve simultaneously:
   \[
   \text{Multiply the first equation by 1.5: } 7.5x + 6y = 600
   \]
   Subtract the second equation:
   \[
   7.5x + 6y - (1.5x + 6y) = 600 - 432 \Rightarrow 6x = 168 \Rightarrow x = 28
   \]
   Substitute $x = 28$ in the first equation:
   \[
   5(28) + 4y = 400 \Rightarrow 140 + 4y = 400 \Rightarrow 4y = 260 \Rightarrow y = 65
   \]
   Intersection point: (28, 65)

### Step 4: Evaluate the Given Options

Given the potential options, let's evaluate:

1. **(0, 0)**
2. **(0, 100)**
3. **(0, 72)**
4. **(28, 65)**
5. **(80, 0)**

From these, the corners of the feasible region should include points that satisfy all constraints.

### Conclusion:
The correct set of corners for the feasible region that satisfies all constraints and is within bounds are:

- **(0, 0)**
- **(0, 72)**
- **(28, 65)**
- **(80, 0)**

So, the correct answer is:

**(0, 0), (0, 72), (28, 65), and (80, 0)**

This corresponds to the third option: 

**(0, 0), (0, 72), (28, 65), and (80, 0)**

Would you like more details or have any questions? 

Here are 5 related questions you might be interested in:
1. How do you graphically represent the feasible region for a linear programming problem?
2. What is the significance of the corners of the feasible region in linear programming?
3. How can you determine the optimal solution in a linear programming problem?
4. How does changing one of the constraints affect the feasible region?
5. What happens if two constraints in a linear programming problem are parallel?

**Tip:** Always verify the intersection points of constraints, as these are crucial in identifying the feasible region in linear programming.

What are the corners of the region bounded by 5x + 4y ≤ 400 which is not part of the feasible region for the following problem? You are in the business of baking bread and cake. A loaf of bread is sold for $2 and a cake for $3. Bread takes 5 minutes and cake takes 4 minutes to bake. Bread takes 0.15 pound of flour dough and cake takes 0.6 pound. You have 6 2/3 hours of labor and 43.2 pounds of flour dough.

Explore how to determine the corners of the feasible region in a linear programming problem involving the baking of bread and cake. This problem includes constraints based on time and dough availability, with a focus on maximizing profit. Ideal for students in Grades 10-12.

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