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GeometryQuestion and Answers: Page 87

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Find the maximum of f(x)=cos x + ((λ tan x−1)/(tan x−λ)) sin x in terms of λ with λ>1 and 0<x<tan^(−1) λ

$F i n d t h e m a x i m u m o f$ $f (x) = \cos x + \frac{λ \tan x - 1}{\tan x - λ} \sin x$ $i n t e r m s o f λ$ $w i t h λ > 1 a n d 0 < x < \tan^{- 1} λ$

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A point source has a distance d to the center of a big sphere with radius R. An other smaller sphere with radius r is placed between the point source and the big sphere. If the distance between the two spheres is constant, say it′s c. Find the maximal shadow area of the small sphere on the surface of the big sphere. Find also the minimal complete shadow of the small sphere on the surface of the big sphere. Assume the small sphere is much smaller than the big sphere such that the big sphere will never completely stay in the shadow of the small sphere.

$A p o i n t s o u r c e h a s a d i s t a n c e d$ $t o t h e c e n t e r o f a b i g s p h e r e w i t h r a d i u s$ $R . A n o t h e r s m a l l e r s p h e r e w i t h r a d i u s$ $r i s p l a c e d b e t w e e n t h e p o i n t s o u r c e$ $a n d t h e b i g s p h e r e . I f t h e d i s t a n c e$ $b e t w e e n t h e t w o s p h e r e s i s c o n s t a n t,$ $s a y {i t}^{'} s c .$ $F i n d t h e m a x i m a l s h a d o w a r e a o f t h e$ $s m a l l s p h e r e o n t h e s u r f a c e o f t h e$ $b i g s p h e r e . F i n d a l s o t h e m i n i m a l$ $c o m p l e t e s h a d o w o f t h e s m a l l s p h e r e$ $o n t h e s u r f a c e o f t h e b i g s p h e r e .$ $A s s u m e t h e s m a l l s p h e r e i s m u c h$ $s m a l l e r t h a n t h e b i g s p h e r e s u c h t h a t$ $t h e b i g s p h e r e w i l l n e v e r c o m p l e t e l y s t a y$ $i n t h e s h a d o w o f t h e s m a l l s p h e r e .$

Question Number 48388 Answers: 2 Comments: 1

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