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

Introduction:
This is a height and distance problem where the observer changes position and the tangent of the angle of elevation is given instead of the angle itself. We use right-angled triangles and the definition of tangent to set up equations involving the height of the tower and the horizontal distances.

Given Data / Assumptions:

• From point P, tan θ = 3/4 for the angle of elevation.
• The observer walks 560 m towards the tower to a new point.
• From the new point, tan θ = 4/3 for the angle of elevation.
• The ground is level and the tower is vertical.
• We must find the height h of the tower.

Concept / Approach:
Let the initial horizontal distance from P to the foot of the tower be d. Then, using tan θ = opposite / adjacent:
From P: h / d = 3/4.From the nearer point: h / (d − 560) = 4/3.These give two equations in h and d. Solving this system yields the height of the tower.

Step-by-Step Solution:
Step 1: From P: h = (3/4) * d.Step 2: From second point: h = (4/3) * (d − 560).Step 3: Equate the two expressions for h:(3/4) * d = (4/3) * (d − 560).Step 4: Multiply both sides by 12 to clear denominators: 9d = 16(d − 560).Step 5: Expand right side: 9d = 16d − 8960.Step 6: Rearrange: 9d − 16d = −8960 ⇒ −7d = −8960 ⇒ d = 1280 m.Step 7: Substitute into h = (3/4)d: h = (3/4) * 1280 = 3 * 320 = 960 m.

Verification / Alternative check:
Check both positions: From P, tan θ = h/d = 960/1280 = 3/4, correct. From the nearer point, distance = 1280 − 560 = 720 m, so tan θ = 960/720 = 4/3, also correct. This verifies that h = 960 m is consistent with both given tangents.

Why Other Options Are Wrong:
Heights like 720 m or 840 m are obtained if you mistakenly use 560 instead of d − 560 or confuse the tangent ratios. 1030 m or 640 m do not satisfy both tangent conditions when checked, so they cannot be the true height.

Common Pitfalls:
Typical errors include misplacing the 560 m term (adding instead of subtracting) or inverting the tangent ratios (using 4/3 in the first case and 3/4 in the second). Careful equation setup and solving are essential.

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