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This problem involves a system in equilibrium, and we are tasked with finding the vertical force \( P \), given that the tension in cable \( AB \) is 60 lbs. The dimensions of the system are clearly provided.

To solve this, we will apply the principle of equilibrium. For a system in equilibrium, both the sum of forces and the sum of moments (about any point) must be zero.

### Key Details:
- Cable \( AB \) has a tension of 60 lbs.
- Dimensions:
  - \( AB = 3 \, \text{in.} \)
  - The horizontal arm has a length of 5 in.
  - The vertical distance from \( D \) to \( B \) is 12 in.

We will:
1. Calculate the moment about point \( D \) to eliminate reaction forces at \( D \).
2. Resolve forces into components.
3. Use the moment equilibrium equation to find the vertical force \( P \).

Let's go through the solution step by step.

### Free Body Diagram (FBD):
- Tension \( T = 60 \, \text{lb} \) in cable \( AB \).
- The horizontal component of tension: \( T_x = T \cos(\theta) \).
- The vertical component of tension: \( T_y = T \sin(\theta) \).

We can calculate \( \theta \) from the geometry of the problem, where:

\[
\tan(\theta) = \frac{3}{5} \quad \Rightarrow \quad \theta = \tan^{-1}\left(\frac{3}{5}\right)
\]

Next, we can calculate the forces and solve for \( P \).

Let me compute this.After calculating the equilibrium condition:

- The vertical force \( P \) is approximately **12.86 lbs**.
- The horizontal component of the tension \( T_x \) in cable \( AB \) is about **51.45 lbs**.
- The vertical component of the tension \( T_y \) in cable \( AB \) is approximately **30.87 lbs**.

Given the answer choices, none of the options directly match this result. It might be possible there is a rounding or interpretation issue in the question, or other forces are being considered.

Would you like more details or a check on any specific part of the process?

Here are 5 follow-up questions:
1. How would the result change if the tension in cable \( AB \) were different?
2. What role does the moment about point \( D \) play in this system?
3. How can we verify the horizontal force equilibrium in this system?
4. What assumptions are we making about the forces at point \( D \)?
5. Could the force \( P \) vary with the angle of the arm?

**Tip:** Always ensure units are consistent when calculating moments or forces.

Assuming that the system below is in equilibrium, if the tension in cable AB is 60 lb, the vertical force P is most nearly:

This problem involves a statics system in equilibrium where the tension in cable AB is 60 lb. The task is to determine the vertical force P using equilibrium conditions and moments. The solution covers trigonometric concepts, moments, and force equilibrium equations.

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