 A boat crosses a 20 meter wide river with a steady current of 1.2 m/s. The boat's speed on the water is 3.4 m/s. At what angle relative to the current must the boat point to move across exactly perpendicular to the river?

To solve this problem, we need to find the angle at which the boat must point relative to the current to move directly across the river, perpendicular to the riverbanks. This is a classic example of vector addition, where the boat’s velocity through the water and the river's current combine to give the boat's resultant velocity.

### Given:
- Width of the river is 20 meters (but this is not necessary for solving the problem).
- Speed of the river's current \( v_r = 1.2 \, \text{m/s} \).
- Speed of the boat relative to the water \( v_b = 3.4 \, \text{m/s} \).

### Step-by-Step Solution:

1. **Set up the velocity components**:
   - The river current flows horizontally with a velocity \( v_r = 1.2 \, \text{m/s} \).
   - The boat's velocity relative to the water is \( v_b = 3.4 \, \text{m/s} \), and we are looking for the angle \( \theta \) at which the boat must point to cancel the effect of the current in the horizontal direction.

2. **Vector components**:
   To move perpendicularly across the river, the horizontal component of the boat's velocity must cancel out the river's current. Thus, the horizontal component of the boat's velocity, which is \( v_b \cdot \sin(\theta) \), should exactly counter the current \( v_r \):
   \[
   v_b \cdot \sin(\theta) = v_r
   \]

3. **Solve for \( \theta \)**:
   \[
   \sin(\theta) = \frac{v_r}{v_b} = \frac{1.2}{3.4}
   \]
   \[
   \sin(\theta) \approx 0.3529
   \]
   Now, find the angle \( \theta \) using the inverse sine function:
   \[
   \theta = \sin^{-1}(0.3529) \approx 20.66^\circ
   \]

### Conclusion:
The boat must point at an angle of approximately \( 20.66^\circ \) **upstream** relative to the current to move directly across the river.

Would you like further clarification or details on any step?

Here are 5 related questions:
1. How would the angle change if the river current increased to 2 m/s?
2. What is the time taken to cross the river at this angle?
3. What would happen if the boat did not compensate for the river's current?
4. How do you compute the resultant velocity of the boat across the river?
5. What would be the boat's angle if the boat’s speed on water was doubled?

**Tip**: Remember, vector addition is key when dealing with combined motion in two dimensions, especially in problems involving velocity and direction!

A boat crosses a 20 meter wide river with a steady current of 1.2 m/s. The boat's speed on the water is 3.4 m/s. At what angle relative to the current must the boat point to move across exactly perpendicular to the river?

Find out how to calculate the angle at which a boat must point to move across a river with a 1.2 m/s current and a boat speed of 3.4 m/s. Learn vector addition and trigonometry to solve this problem, including detailed step-by-step calculations. Suitable for high school students in Grades 10-12.

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