Real Life Problems
The Statue of Liberty
What is the best distance from which to view it?
Two Proposed Solutions
Liberty from the top of the Empire State Building …too far…
From right underneath it - too close… heavily foreshortened
The problem from my calculus book… What is the best distance to view the Statute of Liberty? Work it out…
Problem devolves to maximising angle – what distance?
G
T
B
Obs
GT
B
G Obs
TooClose
Too far
Obs
T
B
G
About right…
46m
46m
46m
46m
46m
46m
X ?
Can get the solution artlessly by simple substitution. But note; answer will not be analytic, and besides is rather inelegant…
G
O
46m
46m
The Analytic Solution involves the differentiation of inverse trigonometric functions… so set up the problem
0
1. Maximise as x moves from 0 to
2. TG/GO = tan(+)
3. (+) = arctan(TG/x), and 4. = arctan(TG/x) – arctan(BG/x)
x
T
B
d/dx = d/dx[(arctan(TG/x) – arctan(BG/x)]
but d/dx(arctan(x)) = (1/(1+x2)) and d/dx(1/x) = -1/x2))
(Must use chain rule as function is in 1/x form)
so that the differentiation devolves to
1 -TG 1 -BG d/dx = ----------------- * ------- - ------------------ * ------
1 + (TG/x)2 x2 1 + (BG/x)2 x2
The differentiation…
Setting d/dx = 0 and substituting statue values we get 0 = [1/(1+(92/x)2) * -92/x2] – [1/(1+(46/x)2) * -46/x2]
0 = -92/(x2 + 8464) + 46/(x2 + 2166)
92x2 + 92*2166 = 46x2 + 46*8464
collecting terms
46x2 = 194672 ; x = sqrt(4232)
=> x 65.05m (at which is 19.47o)
The Calculation…
Realisation:Need a zoom lens from the ferry to see it front-on!
But if you can afford a zoom lens, you don’t have to do the maths…
In fact, you don’t have to do anything if you have the money
Above all – strive for elegance!
For optimal viewing, the formula for the distance D to stand from a statue of height S and a plinth of height P is
D = √(S x P) + (P2))
Hence D = √(46*46) + (462)) ≈ 65.05m
For Nelson’s statue on Trafalgar square, S = 5, P = 49; D ≈ 51
Source: “Why do buses come in threes?” Eastway and Wyndham, ISBN 1-86105-862-4
Eastman and Wyndham give no derivation of the formula, probably because it’s a popular book and not meant to burden the reader with mathematical abstractions… I suppose their formula is derived from planar geometry rather than calculus.
A simpler formula..
Thank you