A man paddles a canoe in a long, straight section of a river. The canoe moves downstream with constant speed 4 m/s relative to the water. The river has a steady current of 1 m/s relative to the bank. The man's hat falls into the river. Eight minutes later, he notices that his hat is missing and immediately turns the canoe around, paddling upriver with the same constant speed of 4 m/s relative to the water. How long does it take the man to row back upriver to reclaim his hat?

Let's solve this step by step.

### Step 1: Determine the distance the hat drifted downstream.
- The hat is moving with the current of the river, which is 1 m/s relative to the bank.
- The man notices his hat missing **8 minutes** after it fell into the water.
  
In these 8 minutes, the hat has been drifting downstream at 1 m/s.

To calculate the distance the hat drifted:
\[
\text{Distance} = \text{speed} \times \text{time} = 1 \, \text{m/s} \times 8 \times 60 \, \text{seconds} = 480 \, \text{meters}.
\]
Thus, the hat is 480 meters downstream from the point where it fell into the water.

### Step 2: Determine the man's speed relative to the bank when paddling upstream.
- The man paddles **upstream** with a speed of 4 m/s relative to the water.
- However, the river's current is flowing **downstream** at 1 m/s.
  
Therefore, the man's effective speed relative to the bank when paddling upstream is:
\[
\text{Man's speed relative to bank} = 4 \, \text{m/s} - 1 \, \text{m/s} = 3 \, \text{m/s}.
\]

### Step 3: Calculate the time it takes for the man to row back to the hat.
The hat is 480 meters downstream from the man when he starts paddling back. His effective speed relative to the bank is 3 m/s. 

To find the time it takes for him to reach the hat:
\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{480 \, \text{meters}}{3 \, \text{m/s}} = 160 \, \text{seconds}.
\]

### Step 4: Convert the time into minutes.
\[
160 \, \text{seconds} = \frac{160}{60} \, \text{minutes} \approx 2.67 \, \text{minutes}.
\]

Thus, it will take the man approximately **2.67 minutes** (or 2 minutes and 40 seconds) to row back upriver and reclaim his hat.

### Summary:
- The man will take about **2.67 minutes** to reach his hat after he turns around.

Would you like more details or have any further questions?

Here are 5 related questions:
1. What if the current was faster, say 2 m/s? How would that change the time?
2. What if the man noticed his hat after 10 minutes instead of 8 minutes?
3. How would the situation change if the river's current speed varied instead of being constant?
4. If the man paddled at a faster speed, say 5 m/s, how much time would he save?
5. How can we model this situation using calculus to account for continuous changes in speed?

**Tip:** Always break down complex motion problems into smaller, manageable steps by separating distances, speeds, and time intervals based on reference frames.

Solve a motion problem where a man paddles a canoe downstream at 4 m/s and notices his hat missing 8 minutes later. Learn how to calculate the time it takes for him to row back upriver, factoring in a river current of 1 m/s. This problem involves relative speed and distance-time relationships, suitable for students in Grades 8-10.

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