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# A point object is 10 cm away from a plane mirror, and the eye of an observer (with pupil diameter 5.0 mm) is 20 cm away. Assuming the eye and the object to be on the same line perpendicular to the mirror surface, find the area of the mirror used in observing the reflection of the point.

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Consider the following graph, with two rays $r$ and $r^\prime$, the two rays hit the mirror at two different points, let the distance between these two points be $x$, the distance between the object and the mirror is $d_0$ and the distance between the eye and the mirror is $d_{ey}$, the incident angle $\theta$ can be written as:

$x=2d_0 \tan(\theta)$

the distance between the pupil and the image is $d_0+d_{ey}$, and let the $R$ be the pupil radius, from the right triangle formed by the image and the eye we can write the angle as:

$\tan(\theta)=\dfrac{R}{d_0+d_{ey}}$

combine these two equation together to get:

$x=\dfrac{2d_0R}{d_0+d_{ey}}$

substitute with the givens to get:

\begin{align*}x&=\dfrac{2(10 \mathrm{~cm})(0.25 \mathrm{~cm})}{10 \mathrm{~cm}+20 \mathrm{~cm}} \\ &=0.167 \mathrm{~cm} \end{align*}

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