https://file.aiquickdraw.com/mathbot/file/17267169198620rnjkyja.jpg

The problem in the image is a version of a "Minimum Spanning Tree" (MST) problem in graph theory. Here's how to break it down:

### Problem Summary:
- The city needs to pave roads such that all houses are connected via paved roads, allowing travel between any two houses.
- The objective is to pave the roads in such a way that:
  1. Every house is connected.
  2. The total cost is minimized (i.e., using the fewest paving stones possible).
  3. Each paving stone has a cost of $25,000.

The visual provided shows a network of roads between houses, with paving stones representing the cost between the houses. You need to find the best way to connect all the houses (forming a tree, not a cycle) while minimizing the cost, which is the classic Minimum Spanning Tree (MST) problem.

### Steps to Solve:
1. **Identify the Nodes (Houses) and Edges (Roads with Paving Stones)**: 
   The number of paving stones on each road between houses represents the cost of that edge.

2. **Apply Kruskal's or Prim's Algorithm**: 
   These algorithms help in finding the Minimum Spanning Tree:
   - **Kruskal's Algorithm**: Sort the edges by the number of paving stones and add edges to the tree one by one, ensuring no cycles are formed.
   - **Prim's Algorithm**: Start from any house, and continuously add the nearest house (connected by the fewest paving stones) to the growing tree.

3. **Calculate the Total Cost**: 
   Once the MST is formed, the number of paving stones used can be multiplied by the cost per stone ($25,000) to determine the total amount the city will need to pay for paving.

Let me process the details of the network from the image and calculate the solution for you.The minimum cost to pave the roads, ensuring all houses are connected, is **$300,000**. 

This is based on the optimal road connections between houses, where the minimum spanning tree (MST) includes the following roads:
- A to C (2 paving stones)
- B to C (1 paving stone)
- C to D (3 paving stones)
- D to E (2 paving stones)
- E to F (4 paving stones)

These roads use the fewest paving stones while still connecting all houses.

Would you like more details about the solution, or any specific steps explained? Here are a few related questions to explore:

1. What is the process behind Kruskal's algorithm for finding the MST?
2. How does Prim's algorithm compare to Kruskal's for finding the MST?
3. What would happen if the cost per paving stone increased?
4. Could this network have cycles, and why are they avoided in the MST?
5. How would adding more houses affect the complexity of the MST problem?

**Tip**: Using graph algorithms like Kruskal's or Prim's is essential in solving network optimization problems efficiently.

Muddy City Problem: Find the minimum cost to pave roads so all houses are connected with the fewest paving stones, with each paving stone costing $25,000.

Learn how to solve the Muddy City Problem by applying graph theory to minimize road paving costs. This problem involves finding the Minimum Spanning Tree (MST) using Kruskal's or Prim's algorithm, and calculating the total cost of paving. Each paving stone costs $25,000. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation