A dentist's chair with a patient in it weighs 2100 N . The output plunger of a hydraulic system begins to lift the chair when the dentist's foot applies a force of 55 N to the plunger and the piston . What is the ratio of the radius of the plunger to the radius of the piston ?

Options

A. 9.3
B. 2.6
C. 6.2
D. 0.64

Answer

To solve the problem, we will use the principle of hydraulic systems, which states that the pressure exerted on both sides of the hydraulic system is equal. We will follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces and Areas**:
- Let \( F_1 \) be the weight of the chair (output force) = 2100 N.
- Let \( F_2 \) be the force applied by the dentist (input force) = 55 N.
- Let \( A_1 \) be the area of the piston (output area).
- Let \( A_2 \) be the area of the plunger (input area).

2. **Set Up the Pressure Equation**:
- According to the hydraulic principle, the pressure on both sides is equal:
\[ \frac{F_1}{A_1} = \frac{F_2}{A_2} \]

3. **Express Areas in Terms of Radii**:
- The area of a circle is given by \( A = \pi r^2 \).
- Therefore, we can express the areas as:
\[ A_1 = \pi r_1^2 \quad \text{and} \quad A_2 = \pi r_2^2 \]

4. **Substitute Areas into the Pressure Equation**:
- Substitute \( A_1 \) and \( A_2 \) into the pressure equation:
\[ \frac{F_1}{\pi r_1^2} = \frac{F_2}{\pi r_2^2} \]
- The \( \pi \) cancels out:
\[ \frac{F_1}{r_1^2} = \frac{F_2}{r_2^2} \]

5. **Rearrange the Equation**:
- Rearranging gives us:
\[ \frac{r_1^2}{r_2^2} = \frac{F_1}{F_2} \]

6. **Substitute the Values**:
- Substitute the values of \( F_1 \) and \( F_2 \):
\[ \frac{r_1^2}{r_2^2} = \frac{2100}{55} \]

7. **Calculate the Ratio of Radii**:
- Taking the square root of both sides:
\[ \frac{r_1}{r_2} = \sqrt{\frac{2100}{55}} \approx \sqrt{38.18} \approx 6.17 \]

8. **Conclusion**:
- The ratio of the radius of the plunger to the radius of the piston is:
\[ \frac{r_1}{r_2} \approx 6.17 \]

### Final Answer:
The ratio of the radius of the plunger to the radius of the piston is approximately 6.17.