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

If this part uses an earlier result, revisit (ii) Optimisation with inverse trigonometry or open all working together.

(iii)

Extreme values on a closed interval

The observer is between 2 m and 15 m from the screen. Find, to the nearest degree, the smallest visual angle.

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Hint

  1. On a closed interval the extreme values occur at stationary points or at the ends.

  2. The only stationary point is the maximum; so the minimum is at an endpoint.

Answer · (iii)

18∘18^\circ, attained at x=15x=15 m.

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Open interactive exploration

On 2 ≤ x ≤ 15 the minimum is at the far end

Switch on the endpoints and the maximum.

Switch on a feature to add it to the sketch and see how it is found. A full-mark sketch shows all of them.

  • θ on the allowed interval
Ideas to try
Which end of the interval gives the smaller angle?