The question is from CAT Geometry - Trigonometry. This question is about Heights and Distances. Our task is to find out the height at which a ballon is flying given the angles made by two observers. Trigonometry is an important topic for CAT Preparation for the CAT Exam. Trigonometric ideas can be completely opposite, as in, some questions test lot of common sense with direction, heights and distances, while others could test you on identities. CAT exam could test one on both these fronts. Make sure you master both in CAT - Trigonometry.

Question 8: Ram and Shyam are 10 km apart. They both see a hot air balloon passing in the sky making an angle of 60° and 30° respectively. What is the height at which the balloon could be flying?

- \\frac{5√3}{2}\\)
- 5√3
- Both A and B
- Can’t be determined

Both A and B

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Case 1 :

tan 30° = \\frac{h}{x}\\)

∴ h = \\frac{x}{√3}\\)

tan 60° = \\frac{h}{(10-x)}\\)

=> h = (10 – x) √3

∴ (10 – x) √3 = \\frac{x}{√3}\\)

=> 30 – 3x = x =)x = \\frac{30}{4}\\)= \\frac {15}{2}\\)

=> ∴h = \\frac{15}{(2√3)})= \\frac {(5√3)}{2}\\)

Case 2 :

tan 60° = \\frac{h}{x}\\) =) h = \\frac{x}{√3}\\)

Also,

tan 30° = \\frac{h}{(10-x)}\\)

=> h = \\frac{(10-x)}{√3}\\) =) 3x – x = 10

=> x = 5

H = 5√3

The question is **"What is the height at which the balloon could be flying"**

Choice C is the correct answer.

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