Nemo Sum walks from point A to point B and then back again, taking the exact same path in the opposite direction. Nemo always walks in one of the cardinal directions, i.e. north, south, east, or west, but not always at the same speed. For reasons known only to higher powers, Nemo always walks at 4 km/h when going North or South; going East, Nemo always walks at 3 km/h; and going West, Nemo always walks at 6 km/h. Nemo’s walk, from the start at point A to the return at point A, lasts exactly 6 hours. 1. How long is the path taken by Nemo from point A to pointB? [3] Note. The question is asking for the length of the path traversed by Nemo, not the distance from A to B.

To solve this problem, we need to figure out how long the total path traveled by Nemo is. Let's break it down step by step.

### Key Information:
- Nemo walks from point A to point B, and then back to point A.
- He walks north or south at 4 km/h, east at 3 km/h, and west at 6 km/h.
- The entire walk lasts 6 hours in total.
- We need to determine the total length of the path, which includes both the trip from point A to point B and back.

Let’s introduce variables for the total time Nemo spends walking in each direction (North, South, East, West):
- \( t_N \): time spent walking north.
- \( t_S \): time spent walking south.
- \( t_E \): time spent walking east.
- \( t_W \): time spent walking west.

Since Nemo takes the same path to point B and back, the time walking north should equal the time walking south (\( t_N = t_S \)) and the time walking east should equal the time walking west (\( t_E = t_W \)).

### Step 1: Total time constraint
The total time of the walk is 6 hours, so the sum of all the time components must add up to 6 hours. However, since the time for each direction is symmetrical, we can focus on just one leg of the journey. Let the time for the one-way trip (from A to B) be \( t_1 \), then:
\[
2t_1 = 6 \text{ hours}, \quad \Rightarrow t_1 = 3 \text{ hours}.
\]
This means the one-way journey from point A to point B takes 3 hours.

### Step 2: Distance formulas for each direction
During the journey, Nemo walks at different speeds depending on the direction:
- The distance walked north \( d_N \) in time \( t_N \) is given by: 
  \[
  d_N = 4t_N \quad (\text{since Nemo walks north at 4 km/h}).
  \]
- The distance walked east \( d_E \) in time \( t_E \) is given by: 
  \[
  d_E = 3t_E \quad (\text{since Nemo walks east at 3 km/h}).
  \]

For the return trip, these distances are reversed, but the time spent going north equals the time spent going south, and the time spent going east equals the time spent going west.

### Step 3: Setting up the equation for the path length
For one leg of the journey (from A to B), the total time spent is 3 hours, so:
\[
t_N + t_E = 3 \text{ hours}.
\]
This is the total time spent walking north and east (or any combination of directions).

The total length of the path Nemo walks is the sum of the distances in each direction for both legs of the journey. Since the walk is symmetrical, the total distance Nemo walks is twice the distance from A to B. That is:
\[
\text{Total distance} = 2(d_N + d_E).
\]

### Step 4: Solving for the distances
Using the speeds:
- \( d_N = 4t_N \) for north and south,
- \( d_E = 3t_E \) for east and west.

From the time equation:
\[
t_N + t_E = 3.
\]

Thus, \( d_N = 4t_N \) and \( d_E = 3(3 - t_N) \).

The total distance is:
\[
\text{Total distance} = 2(4t_N + 3(3 - t_N)) = 2(4t_N + 9 - 3t_N) = 2(t_N + 9).
\]

### Step 5: Maximizing path length
For a valid total time of 6 hours, this equation gives the total path length in terms of \( t_N \). However, to determine the exact result, further context might be needed for minimizing or maximizing travel based on Nemo's speed pattern.

Would you like to explore details for clarification, or get more questions related to the problem?

Nemo Sum walks from point A to point B and then back again, taking the exact same path in the opposite direction. Nemo always walks in one of the cardinal directions, i.e. north, south, east, or west, but not always at the same speed. For reasons known only to higher powers, Nemo always walks at 4 km/h when going North or South; going East, Nemo always walks at 3 km/h; and going West, Nemo always walks at 6 km/h. Nemo’s walk, from the start at point A to the return at point A, lasts exactly 6 hours. 1. How long is the path taken by Nemo from point A to point B?

Solve Nemo's path length calculation problem, where he walks from point A to point B and back, using algebraic equations and speed-time-distance relationships. Learn how to find the total length of the path in this detailed solution, suitable for high school students.

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