A spherical balloon of radius r subtends an angle Φ at the eye of an observer.  If the angle of elevation of its centre is Φ , then find the height of the centre of the balloon .

Asked by pdcavita | 19th Nov, 2017, 02:53: PM

Expert Answer:

 
begin mathsize 16px style Let space straight O space be space the space centre space of space the space balloon space of space radius space straight r space and space straight P space be space the space eye space of space an space observer.
Let space PA comma space PB space be space the space tangents space from space straight P space to space the space balloon.
Then comma space angle APB equals straight alpha
rightwards double arrow angle APO equals angle BPO equals straight alpha over 2
Let space OL space be space perpendicular space from space straight O space on space the space horizontal space PX.
AT space the space centre comma space angle OPL equals straight beta
In space triangle OAP comma
sin space straight alpha over 2 equals OA over OP equals straight r over OP
rightwards double arrow OP equals straight r space cosec space straight alpha over 2
In space triangle OPL comma
sin space straight beta equals OL over OP
rightwards double arrow OL equals OP space sin space straight beta equals straight r space cosec space straight alpha over 2 space sin space straight beta end style

Answered by Rashmi Khot | 30th Nov, 2017, 10:42: AM

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