An aeroplane when 3000 m high passes vertically above another aeroplane at an instant when their angles of elevation at the same observing point are $60^{\circ}$ and $45^{\circ}$ respectively. How many metres higher is the one than the other?

1248 m

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1188 m

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1752 m

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1268 m

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Solution

The correct option is D 1268 m
Here the planes are located at points A and B and the total height from the ground of the highest plane is given as 3000 m.

$begin mathsize 12px style therefore x space plus space h space equals space 3000 m F o r space increment B C D comma space tan space 45 degree space equals space fraction numerator B C over denominator C D end fraction 1 space equals space x over y therefore space x space equals space y space space space space space minus negative negative negative space left parenthesis 1 right parenthesis F o r space increment A C D comma tan space 60 degree space equals space fraction numerator A C over denominator C D end fraction square root of 3 space equals space fraction numerator x space plus space h over denominator y end fraction therefore space square root of 3 space equals space 3000 over y y space equals space fraction numerator 3000 over denominator square root of 3 end fraction space space space space minus negative negative negative left parenthesis 2 right parenthesis U sin g space left parenthesis 1 right parenthesis comma x space equals fraction numerator 3000 over denominator square root of 3 end fraction space therefore space x space equals space fraction numerator 3000 space cross times square root of 3 over denominator 3 end fraction equals 1000 square root of 3 space equals space 1000 space cross times 1.73 space equals space 1730 m end style$

Now distance between the planes = AC - BC = 3000 - x = 3000 - 1730 = 1270m (approx.)

Distance between the planes is 1268 to be more precise.

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