Susan is swimming due west at a speed of 3 meters/second. There is also a strong current flowing S 20° E at a rate of 1 meter/second. Answer the following questions:
Susan's resultant speed is:  Blank 1 Question 4 (Round answer to nearest tenths)
Susan is swimming at a direction angle of:  Blank 2 Question 4   (The angle should be given as the true direction angle rounded to the nearest whole degree)

To determine Susan's resultant speed and direction, we need to use vector addition to combine her swimming velocity and the current's velocity.

### Step 1: Represent the velocities as vectors
- Susan's swimming velocity vector, \( \vec{V}_s \), is due west (270°), with a magnitude of 3 m/s.
  - In component form: \( \vec{V}_s = -3\hat{i} \) m/s (since west is in the negative x-direction).

- The current's velocity vector, \( \vec{V}_c \), is 1 m/s at an angle of S 20° E.
  - This means the current is 20° east of due south (or equivalently, 200° from the positive x-axis).
  - In component form:
    \[
    \vec{V}_c = 1 \cos(200^\circ) \hat{i} + 1 \sin(200^\circ) \hat{j}
    \]
  - Using the trigonometric values:
    \[
    \cos(200^\circ) = \cos(180^\circ + 20^\circ) = -\cos(20^\circ) \approx -0.9397
    \]
    \[
    \sin(200^\circ) = \sin(180^\circ + 20^\circ) = -\sin(20^\circ) \approx -0.3420
    \]
  - Thus:
    \[
    \vec{V}_c \approx 1(-0.9397) \hat{i} + 1(-0.3420) \hat{j} = -0.9397 \hat{i} - 0.3420 \hat{j} \, \text{m/s}
    \]

### Step 2: Add the velocity vectors
- Sum the components of \( \vec{V}_s \) and \( \vec{V}_c \):
  \[
  \vec{V}_r = \vec{V}_s + \vec{V}_c = (-3 \hat{i}) + (-0.9397 \hat{i} - 0.3420 \hat{j}) = (-3.9397 \hat{i} - 0.3420 \hat{j}) \, \text{m/s}
  \]

### Step 3: Determine the resultant speed (magnitude of \( \vec{V}_r \))
- Use the Pythagorean theorem to find the magnitude:
  \[
  |\vec{V}_r| = \sqrt{(-3.9397)^2 + (-0.3420)^2} \approx \sqrt{15.5205 + 0.1160} \approx \sqrt{15.6365} \approx 3.96 \, \text{m/s}
  \]
- Rounded to the nearest tenth, Susan's resultant speed is **4.0 m/s**.

### Step 4: Determine the direction angle
- Calculate the angle θ relative to the negative x-axis using the tangent function:
  \[
  \tan \theta = \frac{|\text{opposite}|}{|\text{adjacent}|} = \frac{0.3420}{3.9397} \approx 0.0868
  \]
  \[
  \theta = \tan^{-1}(0.0868) \approx 4.96^\circ
  \]
- Since this angle is south of the west direction, we subtract it from 270°:
  \[
  \text{True direction angle} = 270° + 4.96° \approx 274.96°
  \]
- Rounded to the nearest whole degree, Susan is swimming at a direction angle of **275°**.

### Summary
1. **Susan's resultant speed**: 4.0 m/s
2. **Susan is swimming at a direction angle of**: 275°

Would you like details on any specific part, or do you have any questions?

### Related Questions:
1. How do we represent velocity vectors in component form?
2. What is the significance of the angle 200° in the current's direction?
3. How can trigonometric functions be used to resolve vector components?
4. Why do we add the components of the vectors instead of their magnitudes directly?
5. What steps are involved in finding the magnitude of a resultant vector?
6. How do we use the tangent function to find the direction angle?
7. Why do we subtract the angle from 270° for the final direction?
8. What is the Pythagorean theorem's role in vector calculations?

### Tip
When working with vectors, always ensure to break them down into their components and handle each direction (i.e., x and y) separately for accurate results.

Learn how to calculate Susan's resultant speed and direction when swimming against a current. This problem involves vector addition and trigonometric calculations, including the Pythagorean theorem and tangent function. Suitable for advanced high school students.

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