provide the full solutions each questions 1. A hydraulic cylinder with a diameter of 10 cm operates at a pressure of 700 kPa. Calculate the force exerted.

2. A hydraulic press with a piston diameter of 0.3 m is supplied with hydraulic fluid at 500 kPa. Calculate the force produced.

3. A hydraulic system exerts a force of 4000 N over an area of 0.05 m². What is the pressure in the system?

4. A hydraulic press applies a force of 12000 N over a piston area of 0.1 m. Calculate the pressure.

5. A hydraulic lift raises a load 3 m while exerting a force of 1500 N. Calculate the work done.

6. A hydraulic cylinder moves a load of 2000 N a distance of 2.5 m. What is the work done?

7. Calculate the energy stored in a hydraulic system with a pressure of 600 kPa and volume of 0.3 m².

8. A hydraulic tank holds oil at a pressure of 400 kPa and has a volume of 0.4 m³. Calculate the energy stored.

9. A hydraulic pump performs 10000 J of work in 20 seconds. What is the power output?

10. A hydraulic system uses 5000 J of energy in 10 seconds. Calculate the power.

11. A hydraulic motor applies a force of 400 N at a radius of 0.25 m. Calculate the torque produced.

12. A hydraulic wrench applies a force of 600 N at a distance of 0.3 m. What is the torque?

Here are the full solutions for each question:

### 1. Force Exerted by a Hydraulic Cylinder
**Given:**
- Diameter = 10 cm = 0.1 m
- Radius \( r = \frac{0.1}{2} = 0.05 \) m
- Pressure \( P = 700 \) kPa = \( 700,000 \) Pa

**Area of the cylinder:**
\[
A = \pi r^2 = \pi (0.05)^2 = \pi (0.0025) \approx 0.00785 \, \text{m}^2
\]

**Force exerted:**
\[
F = P \times A = 700,000 \times 0.00785 \approx 5493.15 \, \text{N}
\]

### 2. Force Produced by a Hydraulic Press
**Given:**
- Diameter = 0.3 m
- Radius \( r = \frac{0.3}{2} = 0.15 \) m
- Pressure \( P = 500 \) kPa = \( 500,000 \) Pa

**Area of the piston:**
\[
A = \pi r^2 = \pi (0.15)^2 = \pi (0.0225) \approx 0.07069 \, \text{m}^2
\]

**Force produced:**
\[
F = P \times A = 500,000 \times 0.07069 \approx 35344.8 \, \text{N}
\]

### 3. Pressure in the Hydraulic System
**Given:**
- Force \( F = 4000 \) N
- Area \( A = 0.05 \) m²

**Pressure:**
\[
P = \frac{F}{A} = \frac{4000}{0.05} = 80000 \, \text{Pa} = 80 \, \text{kPa}
\]

### 4. Pressure from a Hydraulic Press
**Given:**
- Force \( F = 12000 \) N
- Area \( A = 0.1 \) m²

**Pressure:**
\[
P = \frac{F}{A} = \frac{12000}{0.1} = 120000 \, \text{Pa} = 120 \, \text{kPa}
\]

### 5. Work Done by a Hydraulic Lift
**Given:**
- Force \( F = 1500 \) N
- Distance \( d = 3 \) m

**Work done:**
\[
W = F \times d = 1500 \times 3 = 4500 \, \text{J}
\]

### 6. Work Done Moving a Load
**Given:**
- Force \( F = 2000 \) N
- Distance \( d = 2.5 \) m

**Work done:**
\[
W = F \times d = 2000 \times 2.5 = 5000 \, \text{J}
\]

### 7. Energy Stored in a Hydraulic System
**Given:**
- Pressure \( P = 600 \) kPa = \( 600,000 \) Pa
- Volume \( V = 0.3 \) m³

**Energy stored:**
\[
E = P \times V = 600,000 \times 0.3 = 180000 \, \text{J}
\]

### 8. Energy Stored in a Hydraulic Tank
**Given:**
- Pressure \( P = 400 \) kPa = \( 400,000 \) Pa
- Volume \( V = 0.4 \) m³

**Energy stored:**
\[
E = P \times V = 400,000 \times 0.4 = 160000 \, \text{J}
\]

### 9. Power Output of a Hydraulic Pump
**Given:**
- Work \( W = 10000 \) J
- Time \( t = 20 \) s

**Power:**
\[
P = \frac{W}{t} = \frac{10000}{20} = 500 \, \text{W}
\]

### 10. Power of a Hydraulic System
**Given:**
- Energy \( E = 5000 \) J
- Time \( t = 10 \) s

**Power:**
\[
P = \frac{E}{t} = \frac{5000}{10} = 500 \, \text{W}
\]

### 11. Torque Produced by a Hydraulic Motor
**Given:**
- Force \( F = 400 \) N
- Radius \( r = 0.25 \) m

**Torque:**
\[
\tau = F \times r = 400 \times 0.25 = 100 \, \text{Nm}
\]

### 12. Torque from a Hydraulic Wrench
**Given:**
- Force \( F = 600 \) N
- Distance \( d = 0.3 \) m

**Torque:**
\[
\tau = F \times d = 600 \times 0.3 = 180 \, \text{Nm}
\]

---

If you have any questions or need further details on any of these calculations, feel free to ask! Here are some related questions to consider:

1. How would changes in pressure affect the force exerted by the hydraulic cylinder?
2. What is the relationship between work and energy in hydraulic systems?
3. How does torque affect the performance of hydraulic motors?
4. What are the efficiency factors to consider in hydraulic systems?
5. How can you increase the power output of a hydraulic system? 

**Tip:** Always double-check the units when calculating in hydraulic systems to ensure consistency!

Provide the full solutions for each question regarding hydraulic systems and calculations.

Explore detailed solutions to various hydraulic problems involving force, pressure, work done, energy stored, and torque produced. This resource is ideal for students in Grades 10-12 studying hydraulics and fluid mechanics.

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