The human eye has an approximate angular resolution of `phi = 5.8xx10^(-4)` rad and a typical photo printer prints a minimum of 300 dpi (dots per inch, `= 2.54 cm`). Aminimum distance 'z' should a printed page be held so that one doesnot see the indivdual dots is ______ .

Options

A. 14.5 cm
B. 20.5 cm
C. 29.5 cm
D. 28 cm

Explanation

To find the minimum distance \( z \) at which a printed page should be held so that one does not see the individual dots, we can follow these steps:

### Step 1: Calculate the linear distance \( L \) between two consecutive dots
Given that the printer prints at a resolution of 300 dots per inch, we first convert this to centimeters.

1 inch = 2.54 cm, so:
\[ \text{Number of dots per cm} = \frac{300 \text{ dots}}{2.54 \text{ cm}} \approx 118.11 \text{ dots/cm} \]

The linear distance \( L \) between two consecutive dots is:
\[ L = \frac{1 \text{ cm}}{118.11 \text{ dots/cm}} \approx 0.0084 \text{ cm} = 0.84 \times 10^{-2} \text{ cm} \]

### Step 2: Use the angular resolution to find the distance \( z \)
The angular resolution \( \phi \) is given as \( 5.8 \times 10^{-4} \) radians. For small angles, we can use the approximation:
\[ \sin \phi \approx \phi \]

From the geometry of the situation, we have:
\[ \sin \phi = \frac{L}{z} \]

Rearranging this gives:
\[ z = \frac{L}{\phi} \]

### Step 3: Substitute the values into the equation
Now substituting the values of \( L \) and \( \phi \):
\[ z = \frac{0.84 \times 10^{-2} \text{ cm}}{5.8 \times 10^{-4} \text{ rad}} \]

### Step 4: Calculate \( z \)
Performing the calculation:
\[ z \approx \frac{0.84 \times 10^{-2}}{5.8 \times 10^{-4}} \approx 14.48 \text{ cm} \]

Rounding this gives:
\[ z \approx 14.5 \text{ cm} \]

### Final Answer
The minimum distance \( z \) that a printed page should be held so that one does not see the individual dots is approximately **14.5 cm**.

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