A car is connected to a tree by a length of rope tied tightly. A 115 pound woman steps in the middle of the rope causing it to deflect by 15 degrees.

And then what? I don't see a question.

Do they ask for the rope tension force?

no, that's all he posted.

He Goes on to ask what the force applied to the rope is the sum of the x's the sum of the y's the force applied to the car and the force applied to the tree

To solve this problem, we can use the concept of force and the principle of equilibrium. Let's break it down step by step:

1. Draw the free body diagram: Visualize the situation by drawing a diagram showing the forces acting on the system. In this case, we have three forces: the tension in the rope, the weight of the woman, and the reaction force from the tree.

2. Identify the forces and angles: Let's assume the tension in the rope is T, the weight of the woman is W, and the angle of deflection is θ (which is given as 15 degrees).

3. Apply the principle of equilibrium: According to the principle of equilibrium, the sum of all forces and moments acting on an object must be zero.

4. Find the horizontal and vertical components of the forces: Since the deflection angle is given, we can find the horizontal and vertical components of the forces using trigonometry.
- Horizontal component of T: Tcos(θ)
- Vertical component of T: Tsin(θ)
- Vertical component of W: -W (since it acts downwards)

5. Equate the vertical forces: Since the car is in equilibrium, the vertical forces must balance each other. Therefore, we can write the equation as: Tsin(θ) - W = 0.

6. Substitute the known values: The weight of the woman is given as 115 pounds. Substituting this value, we get: Tsin(15) - 115 = 0.

7. Solve for T: Rearrange the equation to solve for T: Tsin(15) = 115. Divide both sides by sin(15) to find T.

8. Use a calculator: Calculate the value of T by substituting sin(15) = 0.2588 in the equation: T ≈ 115 / 0.2588.

9. Compute the tension in the rope: Use a calculator to find the value of T. The approximate tension in the rope is the calculated value of T.

By following these steps, you can determine the tension in the rope when a 115 pound woman steps on it and causes a 15-degree deflection.