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An arc of a bridge is semi-elliptical with major axis horizontal. If the length of the base is 9 meter and the highest part of the bridge is 3 meter from the horizontal then the best approximation of the height of the arch 2 meter from the center of the base is

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a

$\frac{7}{2} m$

b

2m

c

$\frac{11}{4} m$

d

$\frac{8}{3} m$

answer is B.

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Detailed Solution

$L e t t h e e q u a t i o n o f s e m i e l l i p t i c a l b e \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (y > 0) \to (1) I t' s l e n g t h o f m a j o r a x i s = 9 \Rightarrow a = \frac{9}{2} G i v e n l e n g t h o f s e m i m i n o r a x i s (b) = 3 (1) b e c o m e s \frac{4 x^{2}}{81} + \frac{y^{2}}{9} = 1 I f x = 2 t h e n y^{2} = \frac{65}{9} \Rightarrow y = \frac{\sqrt{65}}{3} \Rightarrow y = \frac{8}{3} (a p p r o x i m a t e l y)$

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