Question

The human eye has an approximate angular resolution of $\phi = 5.8 \times {10^{ - 4}}$ rad and typical photo printer prints a minimum of $300$ dpi (dots per inch, $1\,{\text{inch = 2}}{\text{.54cm)}}{\text{.}}$ At what minimal distance z should a printed page be held so that one does not see the individual dots?
(A) $14.5{\text{ cm}}$
(B) $20.5{\text{ cm}}$
(C) $29.5{\text{ cm}}$
(D) ${\text{28 cm}}$

Answer 1

Here we will use the formula to find the distance, first finding the linear distance and the angular resolution and then simplifying the expression placing the given values.

Complete step by step solution:
Given that: Angular resolution of human eye, $\phi = 5.8 \times {10^{ - 4}}rad$
Now, the linear distance between two successive dots in the typical photo printer can be given as –
$l = \dfrac{{2.54}}{{300}}cm$
Simplify the above expression –
$l = 0.84 \times {10^{ - 2}}cm$
At the distance of z cm, the gap distance will subtend an angle and it can be expressed as –
$\phi = \dfrac{l}{z}$
Make the required term as the subject –
$z = \dfrac{l}{\phi }$
Place the values in the above equation –
$z = \dfrac{{0.84 \times {{10}^{ - 2}}}}{{5.8 \times {{10}^{ - 4}}}}$
Simplify the above expression using the laws of power and exponent –
$\Rightarrow z = 14.5cm$
This is the required solution.
Hence, from the given multiple choices the option A is the correct answer.

Additional Information:
Always remember the laws of power and exponent and use it for simplification. The power is used to express mathematical equations in the short form; it is an expression that represents the repeated multiplication of the same factor. For example - $2 \times 2 \times 2$ can be expressed as ${2^3}$ . Here, the number two is called the base and the exponent represents the number of times the base is used as the factor. Remember the seven basic rules of the exponent.

Note :
Always remember the standard formula for the terms mentioned and then frame the new equation finding the correlation between the terms and then place the values. All the units of the placed values should be in the same system of format.