Question

A pinhole camera has the hole a distance$\mathbf{12}\mathit{c}\mathit{m}$from the film plane, which is a rectangle of height $\mathbf{8}\mathbf{.}\mathbf{0}\mathit{c}\mathit{m}$and width $\mathbf{6}\mathbf{.}\mathbf{0}\mathit{c}\mathit{m}$ . How far from a painting of dimensions $\mathbf{50}\mathit{c}\mathit{m}$ by $\mathbf{50}\mathit{c}\mathit{m}$should the camera be placed so as to get the largest complete image possible on the film plane?

Short answer

The distance at which the camera is placed to get a complete image is$100cm$.

Step-by-step solution

The given data:

• Pinhole distance from a film, $d=12cm$
• Height of rectangular film, $h=8cm$
• Width of rectangular film, $w=6cm$
• Painting dimensions,$50cm\times 50cm$$50cm\times 50cm$

Understanding the concept of the geometry

The angle inclination of a line is the angle formed by the intersection of the line and the x-axis. Considering the given geometry as shown below, you can get the required angle of inclination to determine the position at which the camera needs to be placed to get a complete image.

Formula:

The tangent equation of an angle of a right angled-triangle,

$\tan \theta =\frac{Perpendiculer}{Base}$ ….. (i)

Calculation of the distance at which the camera is placed:

1. 
   

From the arrangement shown in the figure, consider the distance$x$of the painting from the pinhole of the camera.

The angle of the inclination can be given using the given data in equation (i) as follows:

$tan\theta =\frac{3\mathrm{cm}}{12\mathrm{cm}}$

role="math" localid="1663048320250" $\begin{array}{l}\theta ={\tan }^{-1}(0.25)\\ =14.03°\end{array}$

Now, the distance at which the came needs to be positioned to get the full image can be calculated using the above value in equation (i) for the bigger triangle as follows:

$tan14.{03}^o=\frac{25\mathrm{cm}}{x}$

$$\begin{gathered} X=\frac{25cm}{\tan (14.03°)} \\ =\frac{25cm}{0.25cm} \\ =100cm \end{gathered}$$

Hence, the required distance is $100\mathrm{cm}$.