Discussion
A small island lies in the middle of a river 100 m wide, and a tall tree stands on the island. P and Q are points directly opposite each other on the two banks. The angles of elevation of the top of the tree from P and Q are \( 30^{\circ} \) and \( 45^{\circ} \) respectively. Find the height of the tree.
Answer: C
P and Q are directly opposite, so the two horizontal distances from the tree add up to the river's full width of 100 m.
With the tree of height \( h \):
\[ d_{30} = \frac{h}{\tan 30^{\circ}} = h\sqrt{3}, \qquad d_{45} = \frac{h}{\tan 45^{\circ}} = h \]
\[ h\sqrt{3} + h = 100 \quad\Longrightarrow\quad h(\sqrt{3}+1) = 100 \]
\[ h = \frac{100}{\sqrt{3}+1} \times \frac{\sqrt{3}-1}{\sqrt{3}-1} = \frac{100(\sqrt{3}-1)}{2} = 50(\sqrt{3}-1) \]
\[ h = 50 \times 0.732 = 36.6 \text{ m} \]
The island is not at the centre: the \( 30^{\circ} \) bank is \( \sqrt{3} \) times further away than the \( 45^{\circ} \) one.
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development