Two vertical trees stand on opposite sides of a straight road, and the distance between the two trees is 400 metres. There is a point on the road between them from which the angles of depression to that point from the tops of the trees are 45° and 60° respectively. If the height of the tree that subtends the 45° angle is 200 metres, what is the height (in metres) of the other tree?

Introduction / Context:
This problem combines the concept of angles of depression with distances along a straight road. Two trees of different heights are located on opposite sides of a road, and a point on the road between them is observed from the top of each tree. The angles of depression to that same point are different, and the height of one tree is known. The total distance between the trees along the road is also given. We need to use right angled triangles and tangent ratios to compute the height of the second tree.

Given Data / Assumptions:

- Two trees stand vertically on opposite sides of a straight road. - The horizontal distance between the two trees is 400 m. - There is a point P on the road between the trees. - Angle of depression from the top of the first tree to point P is 45°. - Angle of depression from the top of the second tree to point P is 60°. - The height of the tree corresponding to the 45° angle is 200 m. - The ground and the road are level.

Concept / Approach:
An angle of depression from the top of a tree to a point on the ground is equal to the angle of elevation from that point to the top of the tree. Therefore, from point P on the road, the angles of elevation to the tops of the trees are 45° and 60°. For the tree of height 200 m with a 45° angle of elevation, the horizontal distance from P to that tree is easy to compute because tan 45° = 1. Once we know that distance, we can find the distance from P to the second tree by subtracting from the total distance of 400 m. With that horizontal distance and the 60° angle of elevation, we can then use tan 60° to find the height of the second tree.

Step-by-Step Solution:
Step 1: Let tree A be the one with height 200 m and angle of depression 45° to point P. Step 2: Let tree B be the unknown height tree with angle of depression 60° to the same point P. Step 3: From point P, the angle of elevation to the top of tree A is 45°, so tan 45° = 200 / dA, where dA is the horizontal distance from P to tree A. Step 4: Since tan 45° = 1, we have 1 = 200 / dA, so dA = 200 m. Step 5: The total distance between the trees is 400 m, so the distance from P to tree B is dB = 400 - dA = 400 - 200 = 200 m. Step 6: From point P, the angle of elevation to the top of tree B is 60°, so tan 60° = hB / dB, where hB is the height of tree B. Step 7: Using tan 60° = √3, we get √3 = hB / 200, so hB = 200√3. Step 8: Therefore, the height of the second tree is 200√3 m.

Verification / Alternative check:
We can check the logic by considering the geometry. If tree A has height 200 m and the horizontal distance to P is also 200 m, then tan of the angle of elevation is 200 / 200 = 1, which corresponds to 45°. For tree B, we use the same horizontal distance 200 m because P is exactly midway between the two trees. With hB = 200√3, we get tan of the angle of elevation as (200√3) / 200 = √3, which corresponds to 60°. Both angles match the given angles of depression, confirming that the distances and heights are consistent.

Why Other Options Are Wrong:
Option A (200 m) would imply both trees have the same height, which would give the same angle of depression from each tree, contradicting the statement that one angle is 45° and the other is 60°. Option C (100√3 m) is smaller than 200√3 and would not produce a 60° angle with the same horizontal distance. Option D (250 m) does not relate neatly to tan 60° or the given distance of 200 m from P, and the resulting tangent would not be √3. Only 200√3 m is consistent with the correct tangent ratio and the geometry of the problem.

Common Pitfalls:
Some learners mistakenly assume that the point on the road is at one end rather than between the trees, leading to incorrect distance calculations. Others forget that angle of depression from the tree equals the angle of elevation from the point on the road and end up misplacing the angles in their diagrams. Another typical error is to use sine or cosine instead of tangent when relating heights to horizontal distances. Drawing a clear diagram and using tan θ = opposite / adjacent for each tree avoids these common mistakes.

Final Answer:
The height of the other tree is 200√3 m, which corresponds to option B.