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Geometry Level 4

A 10 m tall statue is placed on a 10 m high pedestal. How far from the base of the pedestal should one stand to get the best (widest angle) view of the statue? Assume that the eye height is 2 m. Answer in the unit of m.


The answer is 12.

Rajen Kapur by Rajen Kapur , 72, India

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2 solutions

Ossama Ismail
Mar 28, 2019

In the above digram:

t a n ( b ) = 8 x and t a n ( a ) = 18 x tan(b) = \dfrac{8}{x} \\ \text{and} \\ tan(a) = \dfrac{18}{x}

t a n ( a − b ) = t a n ( a ) − t a n ( b ) 1 + t a n ( a ) t a n ( b tan(a-b) = \dfrac{tan(a) - tan(b)}{1 + tan(a) tan(b}

We want to maximiza ( a − b ) (a -b) to get maximum view,

t a n ( a − b ) = 18 x − 8 x 1 + 8.18 x 2 tan(a-b) = \dfrac{ \dfrac{18}{x} - \dfrac{8}{x}}{1 + \dfrac{8.18}{x^2}}

t a n ( a − b ) = 10 x x 2 + 8.18 tan(a-b) = \dfrac{10 x }{x^2 +8.18}

By differentiating with respect to x x to find the maximum. We will get x = 12 x = 12 .

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We can use the maxima-minima relationship here in calculus..

Let x=distance from the base.. then x=sqrt{8*18}=12..

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