From an aeroplane flying above a river at an altitude of 1200 m, it is observed that the angles of depression of opposite points on the two banks of a river are 30^{\circ} and \theta. If the width of the river is 3000 m, then which one of the following is correct?
- A. \theta \lt 30^{\circ}
- B. 30^{\circ} \lt \theta \lt 45^{\circ}
- C. 45^{\circ} \lt \theta \lt 60^{\circ} ✓
- D. 60^{\circ} \lt \theta \lt 90^{\circ}
Correct Answer: C. 45^{\circ} \lt \theta \lt 60^{\circ}
Explanation
The horizontal distance to the first bank is 1200 \cot 30^{\circ} = 1200\sqrt{3} \approx 2078.4 m. The remaining distance to the second bank is 3000 - 2078.4 = 921.6 m. Thus, \cot \theta = \frac{921.6}{1200} \approx 0.768. Since \cot 60^{\circ} \approx 0.577 and \cot 45^{\circ} = 1, the angle \theta lies between 45^{\circ} and 60^{\circ}.
Related questions on Trigonometry
- Two poles are situated 24 m apart and their heights differ by 10 m. What is the distance between their tips?
- If \frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta} = 4, then which one of the following is a value of $(\tan^2 \...
- For 0 < \theta < \frac{\pi}{2}, consider the following : I. $(\tan^4 \theta + \tan^6 \theta)(\cot^4 \theta + \cot^6 \theta) = \sec^2 \the...
- If 3\sin \theta + 4\cos \theta = 5, then what is a value of 4\tan \theta + 3\cot \theta ?
- At a point on level ground, the tangent of the angle of elevation of the top of a tower is found to be \frac{5}{6}. On walking 70 m toward...