A vertical tower stands on the top of a 40 metre high building. From the top and bottom of the tower, the angles of depression of a point on the ground are 60 degrees and 45 degrees respectively. What is the height of the tower in metres?

Introduction / Context:
This is a classic height and distance problem from trigonometry, applied to a real life situation involving a building and a tower. It checks your understanding of angles of depression and your ability to convert these into angles of elevation at the ground point. Using basic tangent ratios in two right triangles that share a common horizontal distance, you can determine the unknown height of the tower.

Given Data / Assumptions:

• Height of the building (up to the bottom of the tower) = 40 metres.
• The tower stands vertically on top of the building.
• Angle of depression from the top of the tower to a point on the ground = 60 degrees.
• Angle of depression from the bottom of the tower (top of building) to the same point = 45 degrees.
• Required: the height of the tower in metres.

Concept / Approach:
The angle of depression from an observer is equal to the angle of elevation from the object to the observer, measured from the horizontal. Let the horizontal distance from the base of the building to the point on the ground be d. Then we can write tan ratios from both the top of the building and the top of the tower, involving d and the corresponding heights. Solving these equations gives the tower height.

Step-by-Step Solution:
Step 1: Let the height of the tower be h metres. Step 2: Let the horizontal distance from the foot of the building to the point on the ground be d metres. Step 3: From the bottom of the tower (top of the 40 metre building), the angle of elevation at the ground point is 45 degrees, so tan 45° = 40/d. Step 4: Since tan 45° = 1, we get 1 = 40/d, which implies d = 40 metres. Step 5: From the top of the tower, the total height above the ground is 40 + h metres, and the angle of elevation at the ground point is 60 degrees. Step 6: Use tan 60° = (40 + h)/d with d = 40. Since tan 60° = sqrt(3), we have sqrt(3) = (40 + h)/40. Step 7: Rearrange to find 40 + h = 40 sqrt(3). Step 8: Solve for h to get h = 40 sqrt(3) - 40 = 40(sqrt(3) - 1).

Verification / Alternative check:
You can verify by substituting h back into the trigonometric relations. The total height from the ground to the top of the tower is 40 + 40(sqrt(3) - 1) = 40 sqrt(3). Dividing this by the horizontal distance 40 gives tan 60° = sqrt(3), confirming the first condition. The height 40 divided by the same distance gives tan 45° = 1, confirming the second condition.

Why Other Options Are Wrong:
20sqrt(3) is too small, as it ignores the given building height and does not satisfy both angle conditions. 30(sqrt(3) + 1) and 50(sqrt(3) - 1) arise from incorrect combinations of the building height and the tangent ratios. 40sqrt(3) represents the total combined height of building plus tower, not just the tower height itself. Only 40(sqrt(3) - 1) matches the derived value for the tower alone.

Common Pitfalls:
Students sometimes confuse angle of depression with angle of elevation and draw incorrect diagrams, leading to wrong tangent ratios. Another common mistake is to treat the building height and tower height separately without recognising that the horizontal distance is common to both right triangles. Carefully labelling the diagram and writing the tangent relations clearly avoids these errors.

Final Answer:
The height of the tower is 40(sqrt(3) - 1) metres.