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Liberty from the top of the Empire State Building …too far…

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.

michael-kemp
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Liberty from the top of the Empire State Building …too far…

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  1. Real Life ProblemsThe Statue of Liberty What is the best distance from which to view it?Two Proposed Solutions

  2. Liberty from the top of the Empire State Building …too far…

  3. From right underneath it - too close… heavily foreshortened

  4. The problem from my calculus book… What is the best distance to view the Statute of Liberty? Work it out…

  5. Problem devolves to maximising angle  – what distance? Too far T 46m  B 46m T G Obs 46m TooClose T About right… B 46m  B  46m 46m G Obs G X ? Obs G Can get the solution artlessly by simple substitution. But note; answer will not be analytic, and besides is rather inelegant…

  6. The Analytic Solution involves the differentiation of inverse trigonometric functions… so set up the problem T 1. Maximise  as x moves from 0 to 2. TG/GO = tan(+)3. (+) = arctan(TG/x), and4.  = arctan(TG/x) – arctan(BG/x) 46m B 46m   O x G  0

  7. The differentiation… 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

  8. The Calculation… Setting d/dx = 0 and substituting statue values we get0 = [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*8464collecting terms46x2 = 194672 ; x = sqrt(4232) => x  65.05m (at which  is 19.47o)

  9. 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!

  10. A simpler formula.. For optimal viewing, the formula for the distance D to stand from a statue of height S and a plinth of height P isD = √(S x P) + (P2))Hence D = √(46*46) + (462))≈ 65.05mFor Nelson’s statue on Trafalgar square, S = 5, P = 49; D ≈ 51Source: “Why do buses come in threes?” Eastway and Wyndham, ISBN 1-86105-862-4Eastman 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.

  11. Thank you

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