From a point P on the ground, the angle of elevation of the top of a vertical tower is such that tan θ = 3/4. After walking 560 metres straight towards the foot of the tower, the tangent of the angle of elevation of the tower becomes 4/3. What is the height (in metres) of the tower?

Introduction / Context:
This is a height and distance problem involving a tower and two different positions of an observer. The key feature is that the tangent of the angle of elevation changes when the observer walks closer to the tower. We can use the definition of tangent in right triangles to set up equations and solve for the height of the tower.

Given Data / Assumptions:

• At the first point P, tan θ₁ = 3/4 for the angle of elevation to the top of the tower.
• The observer walks 560 metres towards the tower in a straight line.
• At the new position, tan θ₂ = 4/3 for the angle of elevation.
• The tower is vertical and the ground is taken to be horizontal.
• We are required to find the height of the tower in metres.

Concept / Approach:
Let the horizontal distance from the initial point P to the foot of the tower be x metres, and let the height of the tower be h metres. At each position, the tangent of the angle of elevation is given by tan θ = opposite side (height) divided by adjacent side (horizontal distance). We will use tan θ₁ = 3/4 and tan θ₂ = 4/3 to form two equations in h and x, then solve for h.

Step-by-Step Solution:
Let x be the initial horizontal distance from P to the foot of the tower. Let h be the height of the tower. At the first position, tan θ₁ = h / x = 3 / 4, so h = (3x) / 4. After walking 560 metres towards the tower, the new distance from the tower is x - 560 metres. At the second position, tan θ₂ = h / (x - 560) = 4 / 3. Substitute h from the first equation: (3x / 4) / (x - 560) = 4 / 3. This gives 3x / (4x - 2240) = 4 / 3. Cross multiply: 3 × 3x = 4 × (4x - 2240). 9x = 16x - 8960, so 7x = 8960 and x = 1280 metres. Now h = (3x) / 4 = (3 × 1280) / 4 = 3840 / 4 = 960 metres.

Verification / Alternative check:
Check the tangents with h = 960 and x = 1280. At the first point, h / x = 960 / 1280 = 3 / 4, which matches tan θ₁. At the second point, horizontal distance is 1280 - 560 = 720, and h / 720 = 960 / 720 = 4 / 3, which matches tan θ₂. So the value is consistent.

Why Other Options Are Wrong:
720: This would give tan θ₁ = 720 / 1280 = 9 / 16, which does not match 3 / 4.
840: Using this height does not satisfy both tangent conditions simultaneously.
1030: This also fails to satisfy the exact fraction values 3 / 4 and 4 / 3 for the given 560 metre movement.

Common Pitfalls:
Common errors include using the wrong sign for the change in distance, forgetting that the second horizontal distance is x - 560 (not x + 560), or trying to use sine or cosine instead of tangent. Another mistake is attempting to plug approximate decimal tangent values instead of using the given simple fractions, which can create unnecessary rounding issues.

Final Answer:
The height of the tower is 960 metres.