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Real Life Problems
The Statue of Liberty
What is the best distance from which to view it?
Two Proposed Solutions
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Liberty from the top of the Empire State Building …too far…
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From right underneath it - too close… heavily foreshortened
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The problem from my calculus book… What is the best distance to view the Statute of Liberty? Work it out…
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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…
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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
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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…
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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…
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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!
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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..
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Thank you