To find the focal length of a convex mirror , a student records the following data:
`{:("Object Pin","Convex Lens","Convex mirror","Image Pin"),(22.2 cm,32.2 cm,45.8 cm,71.2 cm):}`
The focal length of the convex lens is `f_(1)` and that of mirror is `f_(2)` . Then taking index correction to be negligibly small , `f_(1)` and `f_(2)` are close to :

Options

A. `f_(1) = 7.8 cm " " f_(2) = 12.7 cm`
B. `f_(1) = 15.6 cm " " f_(2) = 25.6 cm`
C. `f_(1) = 12.7 cm " " f_(2) = 7.8 cm`
D. `f_(1) = 7.8cm " " f_(2) = 25.4 cm`

Answer

To find the focal length of the convex mirror using the given data, we can follow these steps:

### Step 1: Determine Object and Image Distances for the Convex Lens
- The object pin is at 22.2 cm and the convex lens is at 32.2 cm.
- The object distance \( U_1 \) for the convex lens is calculated as:
\[ U_1 = \text{Convex Lens Position} - \text{Object Pin Position} = 32.2 \, \text{cm} - 22.2 \, \text{cm} = 10 \, \text{cm} \]
Since the object is on the same side as the incoming light, we take it as negative:
\[ U_1 = -10 \, \text{cm} \]

### Step 2: Determine Image Distance for the Convex Lens
- The image pin is at 71.2 cm and the convex lens is at 32.2 cm.
- The image distance \( V_1 \) is calculated as:
\[ V_1 = \text{Image Pin Position} - \text{Convex Lens Position} = 71.2 \, \text{cm} - 32.2 \, \text{cm} = 39 \, \text{cm} \]

### Step 3: Calculate the Focal Length of the Convex Lens
Using the lens formula:
\[ \frac{1}{F_1} = \frac{1}{V_1} - \frac{1}{U_1} \]
Substituting the values:
\[ \frac{1}{F_1} = \frac{1}{39} - \frac{1}{-10} \]
This simplifies to:
\[ \frac{1}{F_1} = \frac{1}{39} + \frac{1}{10} \]
Finding a common denominator (390):
\[ \frac{1}{F_1} = \frac{10}{390} + \frac{39}{390} = \frac{49}{390} \]
Thus,
\[ F_1 = \frac{390}{49} \approx 7.8 \, \text{cm} \]

### Step 4: Determine Radius of Curvature for the Convex Mirror
- The convex mirror is at 45.8 cm and the image pin is at 71.2 cm.
- The radius of curvature \( R \) is calculated as:
\[ R = \text{Image Pin Position} - \text{Convex Mirror Position} = 71.2 \, \text{cm} - 45.8 \, \text{cm} = 25.4 \, \text{cm} \]

### Step 5: Calculate the Focal Length of the Convex Mirror
Using the formula for the focal length of a mirror:
\[ F_2 = \frac{R}{2} \]
Substituting the value of \( R \):
\[ F_2 = \frac{25.4}{2} = 12.7 \, \text{cm} \]

### Final Answer
The focal lengths obtained are:
- Focal length of the convex lens \( F_1 \approx 7.8 \, \text{cm} \)
- Focal length of the convex mirror \( F_2 \approx 12.7 \, \text{cm} \)