A converging lens of focal length `5.0cm` is placed in contact
with a diverging lens of focal length `10.0cm.` Find the combined focal length of
the system.

Explanation

To find the combined focal length of a system of lenses, we can use the lens formula for two lenses in contact:

\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \]

Where:
- \( F \) is the combined focal length of the system.
- \( f_1 \) is the focal length of the first lens (converging lens).
- \( f_2 \) is the focal length of the second lens (diverging lens).

### Step-by-Step Solution:

1. **Identify the Focal Lengths:**
- The focal length of the converging lens \( f_1 = +5.0 \, \text{cm} \).
- The focal length of the diverging lens \( f_2 = -10.0 \, \text{cm} \) (negative because it is a diverging lens).

2. **Apply the Lens Formula:**
\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \]
Substitute the values of \( f_1 \) and \( f_2 \):
\[ \frac{1}{F} = \frac{1}{5.0} + \frac{1}{-10.0} \]

3. **Calculate Each Term:**
- Calculate \( \frac{1}{5.0} = 0.2 \).
- Calculate \( \frac{1}{-10.0} = -0.1 \).

4. **Combine the Terms:**
\[ \frac{1}{F} = 0.2 - 0.1 = 0.1 \]

5. **Find the Combined Focal Length:**
\[ F = \frac{1}{0.1} = 10.0 \, \text{cm} \]

6. **Interpret the Result:**
The positive value of \( F \) indicates that the combined system behaves as a converging lens with a focal length of \( 10.0 \, \text{cm} \).

### Final Answer:
The combined focal length of the system is \( 10.0 \, \text{cm} \).

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