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To find the resultant force on the ocean liner, we need to analyze the two forces exerted by the tugboats. Each tugboat is exerting a force of 400 pounds at an angle of 20 degrees from the x-axis (one at +20 degrees and the other at -20 degrees).

Let's break down each force into its horizontal (\(x\)) and vertical (\(y\)) components, and then combine them to find the resultant force.

### Step 1: Calculate the Components of Each Force
Let \( F_1 = 400 \) pounds at an angle of \( 20^\circ \) to the x-axis (for the lower tugboat) and \( F_2 = 400 \) pounds at an angle of \( -20^\circ \) to the x-axis (for the upper tugboat).

#### Force \( F_1 \):
- **Horizontal component**: \( F_{1x} = 400 \cos(20^\circ) \)
- **Vertical component**: \( F_{1y} = 400 \sin(20^\circ) \)

#### Force \( F_2 \):
- **Horizontal component**: \( F_{2x} = 400 \cos(-20^\circ) = 400 \cos(20^\circ) \) (since cosine is even)
- **Vertical component**: \( F_{2y} = 400 \sin(-20^\circ) = -400 \sin(20^\circ) \) (since sine is odd)

### Step 2: Compute the Values of the Components
Using approximate values:
- \( \cos(20^\circ) \approx 0.9397 \)
- \( \sin(20^\circ) \approx 0.3420 \)

Thus:
- \( F_{1x} = 400 \times 0.9397 = 375.88 \) pounds
- \( F_{1y} = 400 \times 0.3420 = 136.8 \) pounds
- \( F_{2x} = 400 \times 0.9397 = 375.88 \) pounds
- \( F_{2y} = -400 \times 0.3420 = -136.8 \) pounds

### Step 3: Sum the Components
Since the vertical components cancel each other out (\( F_{1y} + F_{2y} = 136.8 - 136.8 = 0 \)), only the horizontal components contribute to the resultant force.

Total horizontal component:
\[
F_x = F_{1x} + F_{2x} = 375.88 + 375.88 = 751.76 \text{ pounds}
\]

Total vertical component:
\[
F_y = F_{1y} + F_{2y} = 136.8 - 136.8 = 0 \text{ pounds}
\]

### Step 4: Calculate the Resultant Force
The resultant force \( F_{\text{resultant}} \) is simply the total horizontal component, as there’s no vertical component.

\[
F_{\text{resultant}} = 751.76 \text{ pounds}
\]

### Answer
The resultant force on the ocean liner is approximately **751.76 pounds** in the horizontal direction (along the x-axis).

Would you like more details on any specific part of this solution, or do you have other questions?

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#### Related Questions:
1. How do we find the resultant if the angles of force application are different?
2. How would we approach this if there were three tugboats exerting force?
3. How does trigonometry help in resolving forces into components?
4. What are other real-world examples of force components being used?
5. How would the solution change if the forces were unequal?

#### Tip:
When resolving forces, always start by breaking each force into perpendicular components along common axes (usually x and y). This simplifies combining multiple forces acting at different angles.

Two tugboats are pushing an ocean liner. Each boat is exerting a force of 400 pounds at an angle of 20 degrees from the x-axis. What is the resultant force on the ocean liner?

Calculate the resultant force exerted on an ocean liner by two tugboats, each applying 400 pounds of force at 20 degrees from the x-axis. This solution involves breaking down forces into horizontal and vertical components, vector addition, and trigonometric calculations. Suitable for students in Grades 10-12.

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