How can we change a camera from F/4 to F/5.6?

Options

A. Increase the aperture to 2 time keeping the focal distance constant.
B. Increase the aperture to `sqrt2` time keeping the focal distance constant.
C. Increase the aperture to `(1)/(2)` time keeping the focal distance constant.
D. Increase the aperture to `(1)/(sqrt2)` time keeping the focal distance constant

Explanation

To change a camera from F/4 to F/5.6, we need to understand how the f-number (or f-stop) relates to the aperture size of the camera lens. The f-number is defined as the focal length of the lens divided by the diameter of the aperture.

### Step-by-Step Solution:

1. **Understand the f-number:**
The f-number (f-stop) is given by the formula:
\[ f = \frac{F}{d} \]
where \( F \) is the focal length of the lens and \( d \) is the diameter of the aperture.

2. **Set up the equations for both f-stops:**
For the first case (F/4):
\[ f_1 = \frac{F}{d_1} = 4 \]
For the second case (F/5.6):
\[ f_2 = \frac{F}{d_2} = 5.6 \]

3. **Express the diameters in terms of focal length:**
From the first equation:
\[ d_1 = \frac{F}{4} \]
From the second equation:
\[ d_2 = \frac{F}{5.6} \]

4. **Divide the two equations to find the relationship between \( d_1 \) and \( d_2 \):**
\[ \frac{d_2}{d_1} = \frac{F/5.6}{F/4} = \frac{4}{5.6} \]
Simplifying this gives:
\[ \frac{d_2}{d_1} = \frac{4}{5.6} = \frac{4}{5.6} \times \frac{10}{10} = \frac{40}{56} = \frac{5}{7} \approx 0.714 \]

5. **Express \( d_2 \) in terms of \( d_1 \):**
Rearranging the above gives:
\[ d_2 = d_1 \times \frac{4}{5.6} = d_1 \times \frac{1}{1.4} \]
This can also be expressed as:
\[ d_2 = \frac{d_1}{\sqrt{2}} \]
(since \( \frac{1}{1.4} \approx \frac{1}{\sqrt{2}} \)).

6. **Conclusion:**
To change the aperture from F/4 to F/5.6, we need to **decrease the diameter of the aperture**. This means we need to **increase the f-number**, which results in a smaller aperture.

### Final Answer:
To change a camera from F/4 to F/5.6, **decrease the aperture size**.

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