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To solve this problem, we need to determine the roller coaster's total displacement by analyzing the different segments of its movement and then using vector addition.

### Step 1: Break Down the Problem
- **Horizontal Movement**: The roller coaster moves 200 feet horizontally.
- **First Incline**: The roller coaster rises 135 feet at an angle of 30.0° above the horizontal.
- **Second Incline**: The roller coaster moves 135 feet at an angle of 40.0° below the horizontal.

### Step 2: Resolve the Inclined Movements into Components
For the inclined movements, we'll resolve the displacement into horizontal (x) and vertical (y) components using trigonometric functions.

#### 1. First Incline (135 ft at 30.0° above the horizontal)
- **Horizontal Component**: \( 135 \times \cos(30^\circ) \)
- **Vertical Component**: \( 135 \times \sin(30^\circ) \)

#### 2. Second Incline (135 ft at 40.0° below the horizontal)
- **Horizontal Component**: \( 135 \times \cos(40^\circ) \)
- **Vertical Component**: \( -135 \times \sin(40^\circ) \)  
  (The negative sign indicates the downward direction.)

### Step 3: Calculate the Total Displacement Components
Add up the horizontal and vertical components.

- **Total Horizontal Displacement (x-component)**:
  \[
  x_{\text{total}} = 200 + 135 \cos(30^\circ) + 135 \cos(40^\circ)
  \]

- **Total Vertical Displacement (y-component)**:
  \[
  y_{\text{total}} = 135 \sin(30^\circ) - 135 \sin(40^\circ)
  \]

### Step 4: Compute the Resultant Displacement
The magnitude of the total displacement is calculated using the Pythagorean theorem:
\[
D = \sqrt{x_{\text{total}}^2 + y_{\text{total}}^2}
\]

### Step 5: Calculate the Direction of the Displacement
The direction (angle θ with respect to the horizontal) can be found using:
\[
\theta = \tan^{-1}\left(\frac{y_{\text{total}}}{x_{\text{total}}}\right)
\]

Let's calculate each of these components.The roller coaster's total displacement from its starting point is approximately **420.77 feet**. The direction of this displacement is approximately **2.63° below the horizontal**.

Would you like more details or have any questions about this solution? Here are some related questions you might explore:

1. How would the displacement change if the angles were different?
2. What is the impact of a steeper incline on the vertical component?
3. How can you calculate the displacement if there were more segments of movement?
4. What if the roller coaster moved in a circular path instead of straight lines?
5. How would you represent this displacement graphically?

**Tip:** When working with vectors, always break them down into their components to simplify calculations, especially when dealing with inclined movements.

A roller coaster moves 200 ft horizontally and then rises 135 ft at an angle of 30.0° above the horizontal. Next, it travels 135 ft at an angle of 40.0° below the horizontal. Use graphical techniques to find the roller coaster’s displacement from its starting point to the end of this movement.

Learn how to calculate the total displacement of a roller coaster using trigonometry and vector addition. This problem involves resolving movements into horizontal and vertical components and applying the Pythagorean Theorem. Suitable for high school students in Grades 9-12.

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