The best seat for the viewing angle

Visual-angle geometryGiven viewing geometry: screen height5m, lower edge1m above the observer eye, distancex, visual angleθ. No optimal distance is supplied.5 m1 mx mθ
Question diagram 1
Visual-angle geometryGiven viewing geometry: screen height5m, lower edge1m above the observer eye, distancex, visual angleθ. No optimal distance is supplied.5 m1 mx mθ

Given screen and sight lines with dimensions5m,1m,x and angleθ.

A movie theatre screen which is 5 m high has its lower edge 1 m above an observer’s eye. The visual angle θ\theta of the observer seated xx m away is as shown in the diagram below.

(i) Show that θ=tan⁡−1(6/x)−tan⁡−1(1/x)\theta=\tan^{-1}(6/x)-\tan^{-1}(1/x). mark

(ii) Find the exact distance the observer should sit to obtain the largest visual angle. [You need not establish that the distance gives the largest visual angle.] marks

(iii) Suppose that the observer is situated between 2 m and 15 m from the screen. Find, to the nearest degree, the smallest visual angle. marks

[2008/RJC/Prelim/I/11(b)]

Total marks: 7

Solution

(i)

Modelling with angles

A screen 5 m high has its lower edge 1 m above an observer's eye. Show that the visual angle is θ=tan⁡−16x−tan⁡−11x\theta=\tan^{-1}\frac6x-\tan^{-1}\frac1x when the observer is xx m away.

Hint

  1. The top of the screen is 6 m above eye level and the bottom is 1 m above.

  2. θ\theta is the difference of the two angles of elevation.

Answer · (i)

θ=arctan⁡(6/x)−arctan⁡(1/x)\theta=\arctan(6/x)-\arctan(1/x).

x>0x>0.