A hydraulic press must be used to press-fit two components together in an assembly process. The system must generate at least 775 lb of force to perform the press fit. The piston at the input cylinder is pushed with a force of 250 lb and has an area of 30 in.². The output can use up to 15 cylinders that are each 30 in.². How many output cylinders must be incorporated into the design?

To determine the number of output cylinders needed, we can apply Pascal’s principle, which states that the pressure applied to a confined fluid is transmitted undiminished throughout the fluid.

### Step 1: Calculate the Input Pressure
The input pressure (\(P\)) is given by:
\[
P = \frac{\text{Force}}{\text{Area}} = \frac{250 \, \text{lb}}{30 \, \text{in}^2} = \frac{25}{3} \, \text{lb/in}^2 \approx 8.33 \, \text{lb/in}^2
\]

### Step 2: Determine the Force Generated by Each Output Cylinder
Each output cylinder has the same area as the input cylinder, which is 30 in². Therefore, the force generated by one output cylinder is:
\[
\text{Force per cylinder} = P \times \text{Area} = 8.33 \, \text{lb/in}^2 \times 30 \, \text{in}^2 = 250 \, \text{lb}
\]

### Step 3: Calculate the Number of Cylinders Required
The total force required is 775 lb. We divide this by the force generated by each cylinder to find the minimum number of cylinders needed:
\[
\text{Number of cylinders} = \frac{775 \, \text{lb}}{250 \, \text{lb per cylinder}} = 3.1
\]

Since we cannot use a fraction of a cylinder, we round up to the nearest whole number. Therefore, **4 output cylinders** are required to meet or exceed the 775 lb force requirement.

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Would you like more details on the calculations or any further explanation? 

Here are some related questions to deepen understanding:

1. What would happen if the input force were increased to 300 lb?
2. How would the calculation change if each output cylinder had an area of 40 in² instead?
3. If only 3 output cylinders were available, what is the maximum force the system could generate?
4. What would be the effect of using a smaller input cylinder area?
5. How does Pascal’s principle apply to this type of hydraulic system?

**Tip:** In hydraulic systems, always check if pressures are evenly distributed and consider safety factors to avoid system overloads.

Discover how to calculate the number of output cylinders needed in a hydraulic press system to achieve a 775 lb force. Learn to apply Pascal's Principle and use formulas for pressure and force in hydraulic systems. Suitable for students in Grades 11-12.

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