The focal length of the lensese of an astronomical telescope are 50 cm and 5 cm. The length of the telescope when the image is formed at the least distance of distinct vision is

Options

A. `45 cm`
B. `55 cm`
C. `(275)/(6) cm`
D. `(325)/(6) cm`

Explanation

To find the length of the astronomical telescope when the image is formed at the least distance of distinct vision, we can follow these steps:

### Step 1: Identify the given values
- Focal length of the objective lens, \( f_O = 50 \, \text{cm} \)
- Focal length of the eyepiece, \( f_E = 5 \, \text{cm} \)
- Least distance of distinct vision, \( D = 25 \, \text{cm} \)

### Step 2: Use the formula for the length of the telescope
The length \( L \) of the telescope when the image is formed at the least distance of distinct vision can be calculated using the formula:
\[ L = f_O + \frac{D \cdot f_E}{D + f_E} \]

### Step 3: Substitute the values into the formula
Substituting the known values into the formula:
\[ L = 50 + \frac{25 \cdot 5}{25 + 5} \]

### Step 4: Calculate the numerator and denominator
Calculate the numerator:
\[ 25 \cdot 5 = 125 \]
Calculate the denominator:
\[ 25 + 5 = 30 \]

### Step 5: Substitute the calculated values back into the equation
Now substituting these values back into the equation:
\[ L = 50 + \frac{125}{30} \]

### Step 6: Simplify the fraction
Now, simplify \( \frac{125}{30} \):
\[ \frac{125}{30} = \frac{25}{6} \]
So, we can rewrite \( L \) as:
\[ L = 50 + \frac{25}{6} \]

### Step 7: Convert 50 to a fraction with a common denominator
Convert 50 to a fraction:
\[ 50 = \frac{300}{6} \]

### Step 8: Add the fractions
Now add the two fractions:
\[ L = \frac{300}{6} + \frac{25}{6} = \frac{325}{6} \]

### Step 9: Final result
Thus, the length of the telescope is:
\[ L = \frac{325}{6} \, \text{cm} \approx 54.17 \, \text{cm} \]