1.At the foot of the mountain the elevation of its summit is 45 degree after asending 1000 m towards the mountain up a slope of 30 degree inclination , the eleation is found out to be 60 degree.find the height of the mountain.
 
2.A round balloon of radius 'a' subtends an angle alpha at the eye of the observer while the angle of elevation of its center is beta. Prove the height of the center of the balloon is a “sin beta cosec alpha /2”.

Asked by tanishqa.rathod | 20th Oct, 2015, 08:38: PM

Expert Answer:

C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
S i n c e space t h e space a n g l e space o f space e l e v a t i o n comma space angle A F D equals 45 degree space a n d space angle C F E equals 30 degree comma space w e space h a v e comma rightwards double arrow angle C F A equals angle A F D minus angle C F E equals 45 degree minus 30 degree equals 15 degree... left parenthesis 1 right parenthesis rightwards double arrow angle E C F equals 90 degree minus 30 degree equals 60 degree  A l s o space angle B C E equals 90 degree space a n d space angle B C A equals 60 degree T h u s comma space angle A C F equals 360 degree minus open parentheses angle B C A plus angle B C E plus angle E C F close parentheses rightwards double arrow angle A C F equals 360 degree minus open parentheses 60 degree plus 90 degree plus 60 degree close parentheses rightwards double arrow angle A C F equals 360 degree minus 210 degree rightwards double arrow angle A C F equals 150 degree  C o n s i d e r space t h e space t r i a n g l e space triangle A C F semicolon B y space a n g l e space s u m space p r o p e r t y comma space w e space h a v e comma angle C A F plus angle C F A plus angle A C F equals 180 degree rightwards double arrow angle C A F plus 15 degree plus 150 degree equals 180 degree space space space space space open square brackets f r o m space left parenthesis 1 right parenthesis close square brackets rightwards double arrow angle C A F equals 180 degree minus 165 degree rightwards double arrow angle C A F equals 15 degree T h u s comma space i n space t r i a n g l e space triangle A C F comma space b a s e space a n g l e s comma space angle C F A space a n d space angle C A F space a r e space e q u a l space t o space 15 degree. H e n c e space triangle A C F space i s space a n space i s o s c e l e s space t r i a n g l e. T h e r e f o r e comma space A C equals C F equals 1000 space space space space open square brackets because g i v e n space C F equals 1000 space m close square brackets  C o n s i d e r space t h e space t r i a n g l e comma space triangle C F E sin 30 degree equals fraction numerator C E over denominator C F end fraction rightwards double arrow 1 half equals fraction numerator C E over denominator 1000 end fraction rightwards double arrow C E equals 500 space m  C o n s i d e r space t h e space t r i a n g l e comma space triangle A B C s i n 60 degree equals fraction numerator A B over denominator A C end fraction rightwards double arrow fraction numerator square root of 3 over denominator 2 end fraction equals fraction numerator A B over denominator 1000 end fraction rightwards double arrow A B equals 500 square root of 3 space m  T h u s comma space t h e space h e i g h t space o f space t h e space m o u n t a i n space i s space A B plus C D equals 500 square root of 3 plus 500 space m.

Answered by Vimala Ramamurthy | 21st Oct, 2015, 12:55: PM