If h is the height or depth (sagitta) of a spherical surfacce and I is the mean distance between the legs of sphermoeter, then radius of curvature R of the surface is

Options

A. `R =(I^2)/(h) + (h)/(2)`
B. `R = (I^2)/(6h) + (h)/(2)`
C. `R =(I^2)/(6h) - (h)/(2)`
D. `R = (h^2)/(6I) + (I)/(2)`

Explanation

To find the radius of curvature \( R \) of a spherical surface given the height \( h \) (sagitta) and the mean distance \( I \) between the legs of a spherometer, we can use the formula:

\[ R = \frac{I^2}{6h} + \frac{h}{2} \]

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let \( h \) be the height or depth (sagitta) of the spherical surface.
- Let \( I \) be the mean distance between the legs of the spherometer.

2. **Understand the Formula**:
- The radius of curvature \( R \) is related to the height \( h \) and the distance \( I \) through the formula:
\[ R = \frac{I^2}{6h} + \frac{h}{2} \]
- This formula is derived from the geometry of the spherical surface and the properties of the spherometer.

3. **Substitute the Values**:
- If you have specific values for \( h \) and \( I \), substitute them into the formula to calculate \( R \).

4. **Calculate Each Term**:
- First, calculate \( \frac{I^2}{6h} \).
- Then calculate \( \frac{h}{2} \).
- Finally, add these two results together to find \( R \).

5. **Final Result**:
- The final value you obtain will be the radius of curvature \( R \) of the spherical surface.

### Example Calculation:
If \( h = 2 \) cm and \( I = 6 \) cm, then:
- Calculate \( \frac{I^2}{6h} = \frac{6^2}{6 \times 2} = \frac{36}{12} = 3 \) cm.
- Calculate \( \frac{h}{2} = \frac{2}{2} = 1 \) cm.
- Thus, \( R = 3 + 1 = 4 \) cm.