https://file.aiquickdraw.com/mathbot/file/1726220669000fpehas97.jpg

The problem provided is about a company renting trucks to transport at least 140 people and 9 tons of goods. There are two types of trucks, **A** and **B**, each with specific constraints and costs:

- **Truck A:**
  - Can carry up to 20 people and 0.6 tons of goods.
  - Rental cost: 4 million VND.
  - Availability: 10 trucks.
  
- **Truck B:**
  - Can carry up to 10 people and 1.5 tons of goods.
  - Rental cost: 3 million VND.
  - Availability: 9 trucks.

The goal is to find the combination of trucks A and B that minimizes the total rental cost while satisfying the transportation requirements.

### Define Variables
Let:
- \( x \) be the number of truck **A** rented.
- \( y \) be the number of truck **B** rented.

### Constraints
1. The total number of people transported:
   \[
   20x + 10y \geq 140
   \]
   
2. The total amount of goods transported:
   \[
   0.6x + 1.5y \geq 9
   \]

3. Availability constraints:
   \[
   x \leq 10 \quad \text{(since only 10 trucks A are available)}
   \]
   \[
   y \leq 9 \quad \text{(since only 9 trucks B are available)}
   \]

4. The objective is to minimize the cost function:
   \[
   \text{Cost} = 4x + 3y
   \]

### Step-by-step Solution:

#### 1. Simplify the constraints:
- From the number of people constraint:
  \[
  20x + 10y \geq 140 \quad \Rightarrow \quad 2x + y \geq 14
  \]

- From the goods constraint:
  \[
  0.6x + 1.5y \geq 9 \quad \Rightarrow \quad 2x + 5y \geq 30
  \]

Now we solve this system of inequalities subject to \( x \leq 10 \) and \( y \leq 9 \), and minimize the cost function \( 4x + 3y \).

### Test possible values for \( x \) and \( y \):

We can try the different combinations provided in the answer choices to find the one with the least cost.

#### Test case A: \( x = 4 \), \( y = 5 \)
- People constraint: \( 2(4) + 5 = 8 + 5 = 13 \), which does **not** satisfy \( 2x + y \geq 14 \).
  - This case is invalid.

#### Test case B: \( x = 5 \), \( y = 6 \)
- People constraint: \( 2(5) + 6 = 10 + 6 = 16 \), which satisfies \( 2x + y \geq 14 \).
- Goods constraint: \( 2(5) + 5(6) = 10 + 30 = 40 \), which satisfies \( 2x + 5y \geq 30 \).
  - Cost: \( 4(5) + 3(6) = 20 + 18 = 38 \).

#### Test case C: \( x = 5 \), \( y = 4 \)
- People constraint: \( 2(5) + 4 = 10 + 4 = 14 \), which satisfies \( 2x + y \geq 14 \).
- Goods constraint: \( 2(5) + 5(4) = 10 + 20 = 30 \), which satisfies \( 2x + 5y \geq 30 \).
  - Cost: \( 4(5) + 3(4) = 20 + 12 = 32 \).

#### Test case D: \( x = 6 \), \( y = 4 \)
- People constraint: \( 2(6) + 4 = 12 + 4 = 16 \), which satisfies \( 2x + y \geq 14 \).
- Goods constraint: \( 2(6) + 5(4) = 12 + 20 = 32 \), which satisfies \( 2x + 5y \geq 30 \).
  - Cost: \( 4(6) + 3(4) = 24 + 12 = 36 \).

### Conclusion:
The minimum cost is obtained for \( x = 5 \) and \( y = 4 \), corresponding to answer **C**.

---

Would you like more details on this solution, or do you have any specific questions? 

Here are 5 related questions you might be interested in:
1. How can we graph inequalities like these to visualize the solution?
2. What methods can be used to solve larger linear programming problems?
3. How would the solution change if more trucks were available?
4. Can we solve this problem using matrix methods?
5. How can we automate this kind of problem-solving using software?

**Tip**: When solving optimization problems, always check the boundaries of your variables to ensure that the constraints are not violated.

A company needs to rent vehicles for transporting at least 140 people and 9 tons of goods with minimal cost. Vehicle A has a capacity of 20 people and 0.6 tons of goods and costs 4 million VND, while Vehicle B can carry 10 people and 1.5 tons, costing 3 million VND. Find the optimal combination of vehicles.

Explore the linear programming approach to minimize transportation costs for a company needing to transport 140 people and 9 tons of goods. Understand how to use constraints and optimization to find the best mix of vehicles.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation