A boat travels due west with a speed of 30 mph. If there is a 7 mph current at N30°W, find the actual speed and direction of the boat.

Let's solve the problem where a boat travels due west at 30 mph, and there is a 7 mph current at N30°W.

### Step 1: Represent the Vectors
- **Boat's velocity vector**: Since the boat is traveling due west, its velocity vector is \( \vec{B} = -30\hat{i} \) (where \( \hat{i} \) represents the west direction).
- **Current's velocity vector**: The current is at N30°W, which means it is 30 degrees west of north. This vector can be broken down into its components:
  \[
  \vec{C} = 7\text{ mph} \cdot (\cos(30^\circ)\hat{j} + \sin(30^\circ)(-\hat{i}))
  \]
  where \( \hat{j} \) represents the north direction.

### Step 2: Calculate the Components of the Current's Velocity
- The northward component (along \( \hat{j} \)):
  \[
  C_y = 7 \cdot \cos(30^\circ) = 7 \cdot \frac{\sqrt{3}}{2} = 7 \cdot 0.866 = 6.062 \text{ mph}
  \]
- The westward component (along \( \hat{i} \)):
  \[
  C_x = 7 \cdot \sin(30^\circ) = 7 \cdot \frac{1}{2} = 3.5 \text{ mph}
  \]

So, the current's velocity vector is:
\[
\vec{C} = -3.5\hat{i} + 6.062\hat{j}
\]

### Step 3: Add the Vectors
The total velocity \( \vec{V} \) of the boat is the sum of the boat's velocity vector and the current's velocity vector:
\[
\vec{V} = \vec{B} + \vec{C} = (-30\hat{i}) + (-3.5\hat{i} + 6.062\hat{j}) = -33.5\hat{i} + 6.062\hat{j}
\]

### Step 4: Calculate the Magnitude of the Resultant Vector
The actual speed of the boat is the magnitude of \( \vec{V} \):
\[
|\vec{V}| = \sqrt{(-33.5)^2 + (6.062)^2} \approx \sqrt{1122.25 + 36.75} \approx \sqrt{1159} \approx 34.05 \text{ mph}
\]

### Step 5: Find the Direction
The direction \( \theta \) relative to the west (measured northward) is given by:
\[
\theta = \tan^{-1}\left(\frac{6.062}{33.5}\right) \approx \tan^{-1}(0.181) \approx 10.3^\circ \text{ north of west}
\]

### Final Answer
The boat's actual speed is approximately **34.05 mph** in a direction of **10.3° north of west**.

Would you like further details or have any other questions?

Here are 5 related questions:
1. How would the boat's actual speed and direction change if the boat traveled at 40 mph instead of 30 mph?
2. What would be the actual speed and direction if the current was stronger, say 10 mph, at the same angle?
3. How does changing the direction of the current to N30°E affect the boat's speed and direction?
4. What if the boat wanted to maintain a due west course despite the current; what adjustments would need to be made?
5. How would the boat's speed and direction change if the current direction was exactly north?

**Tip:** To solve vector problems, drawing a diagram to visualize the components and angles can significantly aid in understanding and solving the problem.

Learn how to calculate the boat's actual speed and direction when traveling westward with a current at N30°W. This problem involves vector addition, velocity components, and trigonometric calculations. Suitable for advanced high school students.

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