Page 206 - Calculus Demystified

Transcendental Functions
CHAPTER 6
6.6.6 AN EXAMPLE INVOLVING INVERSE 193
TRIGONOMETRIC FUNCTIONS
EXAMPLE 6.42
Hypatia isviewing a ten-foot-long tapestry that ishung lengthwise on a
wall.The bottom end of the tapestry is two feet above her eye level. At what
distance should she stand from the tapestry in order to obtain the most
favorable view?

SOLUTION
For the purposes of this problem, view A is considered more favorable than
view B if it provides a greater sweep for the eyes. In other words, form the
triangle with vertices (i) the eye of the viewer, (ii) the top of the tapestry, and
(iii) the bottom of the tapestry (Fig. 6.27). Angle α is the angle at the eye of
the viewer. We want the viewer to choose her position so that the angle α at the
eye of the viewer is maximized.

10 ft

2 ft

x ft

Fig. 6.27

The figure shows a mathematical model for the problem. The angle α is the
angle θ less the angle ψ. Thus we have

α = θ − ψ = Cot −1 (x/12) − Cot −1 (x/2).

Notice that when the viewer is standing with her face against the wall then
θ = ψ = π/2 so that α = 0.Also when the viewer is far from the tapestry then
θ − α is quite small. So the maximum value for α will occur for some finite,
positive value of x. That value can be found by differentiating α with respect
to x, setting the derivative equal to zero, and solving for x.