From a point P, the angle of elevation of a vertical tower is such that its tangent is 3/4. After walking 560 metres straight towards the tower along level ground, the tangent of the angle of elevation becomes 4/3. What is the height of the tower in metres?

Introduction / Context:
This problem examines how the angle of elevation of a tower changes as an observer moves closer to it. Instead of giving the actual angles, the question provides the values of their tangents, which is equivalent information. Using the tangent ratio definition, we can set up equations in terms of the tower height and horizontal distances, and then solve them to find the height of the tower.

Given Data / Assumptions:

• From point P, tan of the angle of elevation of the tower is 3/4.
• After walking 560 metres directly towards the tower, tan of the new angle of elevation is 4/3.
• The tower is vertical and stands on level ground.
• The observer walks along a straight horizontal line towards the tower.
• We ignore any height of the observer and assume measurements from ground level.

Concept / Approach:
If the height of the tower is h metres and the initial horizontal distance from point P is x metres, then tan θ1 = h / x. Given tan θ1 = 3/4, we obtain a relation between h and x. After walking 560 metres closer, the distance becomes x − 560 metres, and tan θ2 = h / (x − 560) is given as 4/3. These two equations in terms of h and x can be solved simultaneously to find x and then h.

Step-by-Step Solution:
Step 1: Let h be the height of the tower and x be the initial horizontal distance from point P. Step 2: From the first position, tan θ1 = h / x = 3 / 4, so h = (3/4)x. Step 3: From the second position, the distance to the tower is x − 560. Here tan θ2 = h / (x − 560) = 4 / 3. Step 4: Substitute h = (3/4)x into the second relation: (3/4)x / (x − 560) = 4 / 3. Step 5: Simplify: (3x) / (4(x − 560)) = 4 / 3. Cross multiply to get 9x = 16(x − 560). Step 6: Expand and solve: 9x = 16x − 8960, so 7x = 8960 and x = 1280 metres. Step 7: Substitute back to find h: h = (3/4)x = (3/4) × 1280 = 960 metres.

Verification / Alternative check:
Verify both tangents. Initial distance 1280 metres with height 960 metres gives tan θ1 = 960 / 1280 = 3 / 4, correct. After walking 560 metres, the distance becomes 1280 − 560 = 720 metres, so tan θ2 = 960 / 720 = 4 / 3, which matches the given value. This confirms that h = 960 metres is consistent with all data.

Why Other Options Are Wrong:
If height were 720 metres, the first tangent would be 720 / 1280 which is not 3 / 4. A height of 840 metres or 1030 metres fails to produce the correct ratio 4 / 3 for the second position when the displacement is 560 metres. Option 880 is another trial value that does not satisfy both tangent conditions together. Only 960 metres satisfies both simultaneously.

Common Pitfalls:
Students sometimes forget that the new distance from the tower is x minus 560 and not x plus 560. Another frequent error is in cross multiplication or simplifying the resulting equation. It is also easy to misinterpret tangents and accidentally invert the ratios, writing x / h instead of h / x. Careful algebra and clear understanding of the tangent definition help avoid these mistakes.

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