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To solve this problem, we need to determine the maximum force \( P \) that can be safely applied at point \( D \) such that the reaction at point \( C \) does not exceed 250 lb.

### Step-by-Step Solution:

1. **Understanding the Problem:**
   - The lever \( BCD \) is hinged at \( C \) and connected to a control rod at \( B \).
   - The lever can pivot about point \( C \), and a force \( P \) is applied at point \( D \).
   - We need to determine the maximum value of \( P \) such that the reaction force at \( C \) is limited to 250 lb.

2. **Free Body Diagram (FBD) and Forces:**
   - At hinge \( C \), there will be a reaction force with both horizontal (\( C_x \)) and vertical (\( C_y \)) components.
   - Force \( P \) is applied at \( D \) in a horizontal direction.
   - The control rod at \( B \) exerts a force, but since we are interested in finding the maximum \( P \), we focus on moments about point \( C \).

3. **Taking Moments about Point \( C \):**
   - The moment arm for the force \( P \) at \( D \) is 7.5 in (vertical distance from \( C \) to \( D \)).
   - The moment arm for the force at \( B \) is 3 in (vertical distance from \( C \) to \( B \)).

   The sum of moments about point \( C \) is given by:
   \[
   M_C = P \times 7.5 - F_B \times 3 = 0
   \]

   Where \( F_B \) is the horizontal force in the control rod at \( B \).

   Solving for \( F_B \), we have:
   \[
   F_B = \frac{P \times 7.5}{3} = 2.5P
   \]

4. **Calculating the Reaction at Point \( C \):**
   - The vertical reaction component \( C_y = 0 \) since there are no vertical forces.
   - The horizontal reaction component \( C_x \) will balance both \( F_B \) and \( P \):
   
   \[
   C_x = F_B + P = 2.5P + P = 3.5P
   \]

   The resultant reaction force at \( C \) is:
   \[
   R_C = C_x = 3.5P
   \]

   Since the maximum allowable value for \( R_C \) is 250 lb:
   \[
   3.5P = 250
   \]

   Solving for \( P \):
   \[
   P = \frac{250}{3.5} \approx 71.43 \text{ lb}
   \]

### Conclusion:
The maximum force \( P \) that can be safely applied at \( D \) is **71.43 lb**.

Would you like more details on this problem or have any further questions?

### Related Questions to Explore:
1. How do you calculate moments in mechanics problems?
2. What is the significance of the reaction forces at a hinge in a static problem?
3. How do different forces affect the equilibrium of a system?
4. How can you use trigonometry to solve mechanics problems involving angles?
5. What are the conditions of equilibrium in two-dimensional force systems?

### Tip:
When solving problems involving moments, always choose a point where the maximum number of unknown forces intersect to simplify the calculations.

The lever BCD is hinged at C and attached to a control rod at B. Determine the maximum force P that can be safely applied at D if the maximum allowable value of the reaction at C is 250 lb.

Find the maximum force P that can be applied at D on a lever BCD, hinged at C, with a control rod at B. The maximum allowable reaction at C is 250 lb. Learn how to calculate moments and reaction forces for static equilibrium problems in engineering.

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