Question

a point object is 10 cm away from a plane mirror while the eye of an observer(pupil diameter is 5.0 mm) is 28 cm a way assuming both eye and the point to be on the same line perpendicular to the surface find the area of the mirror used in observing the reflection of the point

Answer 1

1.37 mm²

Step-by-step explanation:

From the image attached below:

Let's take a look at the two rays r and r' hitting the same mirror from two different positions.

Let x be the distance between these rays.

`d_o =` distance between object as well as the mirror

`d_(eye)` = distance between mirror as well as the eye

Thus, the formula for determining the distance between these rays can be expressed as:

`x = 2d_o tan \theta`

where; the distance between the eye of the observer and the image is:

`s = d_o + d_(eye)`

Then, the tangent of the angle θ is:

`tan \theta = (R)/(d_o+d_(eye))`

replacing
`tan \theta = (R)/(d_o+d_(eye))` into
`x = 2d_o tan \theta`, we have:

`x = 2d_o \Big( (R)/(d_o+d_(eye))\Big)`

`x = 2(10) \Big( (0.25)/(10+28)\Big)`

`x = 20\Big( (0.25)/(38)\Big) cm`

x = (0.13157 × 10) mm

x = 1.32 mm

Finally, the area A = π r²

`A = \pi((x)/(2))^2`

`A = \pi((1.32)/(2))^2`

A = 1.37 mm²