Question 22. On the occasion of the Mid-Autumn Festival, the onion factory wants to produce two types of cakes: pies and cakes.
flexible. To produce these two types of cakes, the factory needs: sugar, flour, eggs, pumpkin jam, sausage... The factory
imported 600kg of flour and 240kg of sugar, other ingredients always met the quantity required.
karma needs. Each cake requires 120g of flour and 60g of sugar. Each sticky cake requires 160g of flour and 40g of sugar. According to market surveys, the amount of sticky rice cakes consumed does not exceed three times the amount of baked cakes and the factory's products are always consumed. Each cake makes a profit of 8,000 VND. Each sticky rice cake earns 6,000 VND. Make a production plan for the factory to meet market demand; ensure that the amount of flour and sugar does not exceed the amount that the factory has prepared and collected
get the highest profit.

To solve this problem, we can use linear programming. We'll define the variables and constraints, then formulate the objective function to maximize profit.

### Variables:
- Let \( x \) be the number of cakes produced.
- Let \( y \) be the number of sticky rice cakes produced.

### Objective Function:
We want to maximize profit:
\[
P = 8000x + 6000y
\]

### Constraints:
1. **Flour constraint**:
   \[
   120x + 160y \leq 600000 \quad (\text{since } 600 \text{ kg} = 600000 \text{ g})
   \]
   
2. **Sugar constraint**:
   \[
   60x + 40y \leq 240000 \quad (\text{since } 240 \text{ kg} = 240000 \text{ g})
   \]

3. **Market demand constraint** (sticky rice cakes do not exceed three times the amount of baked cakes):
   \[
   y \leq 3x
   \]

4. **Non-negativity constraints**:
   \[
   x \geq 0, \quad y \geq 0
   \]

### Steps to Solve:
1. **Graph the constraints**: Plot the inequalities on a graph.
2. **Find the feasible region**: Identify the region that satisfies all constraints.
3. **Identify corner points**: Evaluate the objective function at each vertex of the feasible region.
4. **Determine maximum profit**: Choose the vertex with the highest profit.

### Step-by-Step Solution:

1. **Flour constraint**: 
   \[
   120x + 160y = 600000
   \]
   Simplifying gives \( 3x + 4y = 15000 \).

2. **Sugar constraint**: 
   \[
   60x + 40y = 240000
   \]
   Simplifying gives \( 3x + 2y = 12000 \).

3. **Market constraint**:
   \[
   y = 3x
   \]

### Finding Intersections:
Now, we find the intersection points of these lines.

1. **Intersection of Flour and Sugar**:
   - Solve:
     \[
     3x + 4y = 15000
     \]
     \[
     3x + 2y = 12000
     \]
   Subtract the second equation from the first:
   \[
   2y = 3000 \Rightarrow y = 1500
   \]
   Substitute \( y \) back to find \( x \):
   \[
   3x + 2(1500) = 12000 \Rightarrow 3x + 3000 = 12000 \Rightarrow 3x = 9000 \Rightarrow x = 3000
   \]
   Point: \( (3000, 1500) \)

2. **Intersection of Flour and Market Demand**:
   - Substitute \( y = 3x \) into the flour constraint:
     \[
     120x + 160(3x) = 600000 \Rightarrow 120x + 480x = 600000 \Rightarrow 600x = 600000 \Rightarrow x = 1000
     \]
     \[
     y = 3(1000) = 3000
     \]
   Point: \( (1000, 3000) \)

3. **Intersection of Sugar and Market Demand**:
   - Substitute \( y = 3x \) into the sugar constraint:
     \[
     60x + 40(3x) = 240000 \Rightarrow 60x + 120x = 240000 \Rightarrow 180x = 240000 \Rightarrow x = 1333.33
     \]
     \[
     y = 3(1333.33) = 4000
     \]
   Point: \( (1333.33, 4000) \)

### Evaluate Profit:
1. For \( (3000, 1500) \):
   \[
   P = 8000(3000) + 6000(1500) = 24000000 + 9000000 = 33000000
   \]

2. For \( (1000, 3000) \):
   \[
   P = 8000(1000) + 6000(3000) = 8000000 + 18000000 = 26000000
   \]

3. For \( (1333.33, 4000) \):
   \[
   P = 8000(1333.33) + 6000(4000) \approx 10666664 + 24000000 = 34666664
   \]

### Conclusion:
The optimal production plan to maximize profit is approximately:
- **Cakes**: 1333
- **Sticky Rice Cakes**: 4000

This will yield the highest profit of about **34,666,664 VND**.

If you have any questions or need further details, feel free to ask! Here are some related questions you might consider:
1. What if the availability of flour changes?
2. How would the profit change with different prices for the cakes?
3. What are the potential impacts of increasing the market demand for sticky rice cakes?
4. How could the production process be optimized further?
5. What other factors should the factory consider beyond just flour and sugar? 

**Tip**: Always analyze the sensitivity of your solution to changes in constraints or objective coefficients.

On the occasion of the Mid-Autumn Festival, the onion factory wants to produce two types of cakes: pies and cakes. To produce these two types of cakes, the factory needs: sugar, flour, eggs, pumpkin jam, sausage... The factory imported 600kg of flour and 240kg of sugar, other ingredients always met the quantity required. Each cake requires 120g of flour and 60g of sugar. Each sticky cake requires 160g of flour and 40g of sugar. According to market surveys, the amount of sticky rice cakes consumed does not exceed three times the amount of baked cakes and the factory's products are always consumed. Each cake makes a profit of 8,000 VND. Each sticky rice cake earns 6,000 VND. Make a production plan for the factory to meet market demand; ensure that the amount of flour and sugar does not exceed the amount that the factory has prepared and collected get the highest profit.

Learn how to create an optimal production plan for an onion factory producing cakes and sticky rice cakes to maximize profit. Using 600 kg of flour and 240 kg of sugar, the solution involves linear programming constraints on flour and sugar while respecting market demand. Each cake yields 8,000 VND profit, while each sticky rice cake earns 6,000 VND. Ideal for students in Grades 11-12 or introductory college-level math.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation