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.