The length of a telescope is 36 cm . The focal lengths of its lenses can be

Options

A. a) 30 cm, 6cm
B. b) `-30cm, -6 cm`
C. c) `30cm, -6cm`
D. d) `-30cm, 6cm`

Answer

To solve the problem regarding the focal lengths of the lenses in a telescope with a total length of 36 cm, we can follow these steps:

### Step 1: Understand the Components of a Telescope
A telescope typically consists of two lenses: the objective lens and the eyepiece (ocular) lens. The distance between these two lenses is equal to the length of the telescope.

### Step 2: Define the Focal Lengths
Let:
- \( f_o \) = focal length of the objective lens
- \( f_e \) = focal length of the eyepiece lens

The total length of the telescope can be expressed as:
\[ L = f_o + f_e \]

### Step 3: Substitute the Given Length
From the problem, we know that the length of the telescope \( L \) is 36 cm. Therefore, we can write:
\[ f_o + f_e = 36 \, \text{cm} \]

### Step 4: Analyze the Options
The options given for the focal lengths are:
- A) \( f_o = 30 \, \text{cm}, f_e = 6 \, \text{cm} \)
- B) \( f_o = 30 \, \text{cm}, f_e = 6 \, \text{cm} \)
- C) \( f_o = 30 \, \text{cm}, f_e = 6 \, \text{cm} \)
- D) \( f_o = 30 \, \text{cm}, f_e = 6 \, \text{cm} \)

### Step 5: Check the Validity of Each Option
Since all options are the same, we can check if they satisfy the equation \( f_o + f_e = 36 \):
- For option A: \( 30 \, \text{cm} + 6 \, \text{cm} = 36 \, \text{cm} \) (valid)
- For option B: \( 30 \, \text{cm} + 6 \, \text{cm} = 36 \, \text{cm} \) (valid)
- For option C: \( 30 \, \text{cm} + 6 \, \text{cm} = 36 \, \text{cm} \) (valid)
- For option D: \( 30 \, \text{cm} + 6 \, \text{cm} = 36 \, \text{cm} \) (valid)

### Step 6: Conclusion
Since all options satisfy the equation, we conclude that the focal lengths of the lenses can indeed be \( f_o = 30 \, \text{cm} \) and \( f_e = 6 \, \text{cm} \). Therefore, the correct answer is:
**A) \( f_o = 30 \, \text{cm}, f_e = 6 \, \text{cm} \)**.