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 (i) Modelling with angles or open all working together.

(ii)

Optimisation with inverse trigonometry

Find the exact distance the observer should sit to obtain the largest visual angle.

Explore this idea

Hint

  1. Use ddxtan⁡−1u=u′1+u2\frac d{dx}\tan^{-1}u=\frac{u'}{1+u^2}.

  2. Set dθdx=0\frac{d\theta}{dx}=0 and solve.

Answer · (ii)

x=6x=\sqrt6 m.

Explore the idea

Open interactive exploration

The visual angle against distance (in degrees)

The angle rises to a maximum at x = √6, then falls off as the observer sits further back.

Position along the curvex = 8
Try an example
Test a critical position

Buttons use full-precision reviewed positions; displayed coordinates are rounded.

f(x)
29.74488
Gradient f′(x)
-2.55627
negative

Beyond x=√6 the angle decreases: its gradient is negative.

Selected values are within the original drawing windows.

  • θ in degrees
  • Tangent
  • Dθ/dx, and where θ increases
  • Join P to Q; the dashed legs show Δx and Δy
Key idea

Sitting very close or very far both give small angles; the best seat is between, at x=6x=\sqrt6.

Ideas to try
Find the seat where the tangent is flat.