 •••  A child in danger of drowning in a river is being carried downstream by a current that flows uniformly with a speed of 2.0 m/s. The child is 200 m from the shore and 1500 m upstream of the boat dock from which the rescue team sets out. If their boat speed is 8.0 m/s with respect to the water, at what angle from the shore should the pilot leave the shore to go directly to the child?

To solve this problem, we'll need to use concepts from relative velocity and vector decomposition. The rescue team's boat must account for both the current of the river and its own speed to travel directly toward the child.

### Step-by-step solution:

1. **Determine the relative motion:**

   The boat is moving at a speed of 8.0 m/s relative to the water. The current is moving at a speed of 2.0 m/s downstream, parallel to the shore. To ensure that the boat goes directly toward the child, the boat's velocity in the downstream direction must cancel out the velocity of the current.

   Let’s define the following:
   - \( v_b = 8.0 \, \text{m/s} \) (boat speed relative to the water),
   - \( v_c = 2.0 \, \text{m/s} \) (speed of the river current downstream),
   - \( \theta \) is the angle the boat must make with the shore to travel directly toward the child.

2. **Break the boat's velocity into components:**

   The boat’s velocity vector can be decomposed into two components:
   - The horizontal component (parallel to the current): \( v_b \cos \theta \),
   - The vertical component (perpendicular to the current, toward the child): \( v_b \sin \theta \).

   The horizontal component of the boat's velocity must cancel out the velocity of the current. Therefore, we set:
   \[
   v_b \cos \theta = v_c
   \]
   Substituting the values:
   \[
   8.0 \cos \theta = 2.0
   \]
   Solving for \( \cos \theta \):
   \[
   \cos \theta = \frac{2.0}{8.0} = 0.25
   \]

3. **Find the angle \( \theta \):**
   \[
   \theta = \cos^{-1}(0.25)
   \]
   Using a calculator:
   \[
   \theta \approx 75.5^\circ
   \]

   Therefore, the pilot must steer the boat at an angle of approximately **75.5 degrees** from the shore to go directly toward the child.

### Summary:
- The angle \( \theta \) from the shore that the pilot should steer the boat is **75.5 degrees**.

Would you like further details or have any other questions?

Here are 5 related questions to expand on this problem:
1. How long will it take for the boat to reach the child?
2. What would happen if the river current were faster?
3. How does the angle change if the boat

A child in danger of drowning in a river is being carried downstream by a current that flows uniformly with a speed of 2.0 m/s. The child is 200 m from the shore and 1500 m upstream of the boat dock from which the rescue team sets out. If their boat speed is 8.0 m/s with respect to the water, at what angle from the shore should the pilot leave the shore to go directly to the child?

Learn how to determine the angle at which a rescue boat must travel to save a child being carried downstream by a river current. The problem involves the use of vector decomposition and relative velocity concepts. The boat's speed is 8.0 m/s, the river current is 2.0 m/s, and the child is 200 m from the shore and 1500 m upstream. Suitable for students in Grades 11-12.

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