Problem Analysis:
Let's solve this problem step by step. We are given the following information:
- The angle of elevation of the tower from point P is such that its tangent is 3/4.
- On walking 560 meters towards the tower, the tangent of the angle of elevation becomes 4/3.

We need to find the height of the tower. To solve this problem, we can use trigonometric ratios and apply the concept of similar triangles.

Step-by-Step Solution:
1. Let's assume the height of the tower is H meters.
2. From point P, the angle of elevation of the tower is such that its tangent is 3/4. This means we can write the trigonometric equation: tan(theta) = 3/4, where theta is the angle of elevation.
3. Recall that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the opposite side is the height of the tower (H) and the adjacent side is the distance between the point P and the tower.
4. Using the given tangent ratio, we can write: H / x = 3/4, where x is the distance between point P and the tower.
5. On walking 560 meters towards the tower, the tangent of the angle of elevation becomes 4/3. This means the new distance between point P and the tower is (x - 560) meters.
6. We can write a similar trigonometric equation using the new tangent ratio: H / (x - 560) = 4/3.
7. Now we have two equations:
- H / x = 3/4 (Equation 1)
- H / (x - 560) = 4/3 (Equation 2)
8. We can solve these equations simultaneously to find the values of H and x.
9. Multiply Equation 1 by 4 and Equation 2 by 3 to eliminate the fractions:
- 4(H / x) = 4 * (3/4)
- 3(H / (x - 560)) = 3 * (4/3)
10. Simplifying the equations, we get:
- H / x = 3
- H / (x - 560) = 4
11. Cross-multiplying, we get:
- H = 3x
- H = 4(x - 560)
12. Substituting the value of H from the first equation into the second equation, we get:
- 3x = 4(x - 560)
- 3x = 4x - 2240
- x = 2240
13. Now that we have the value of x, we can substitute it back into the first equation to find the value of H:
- H = 3x
- H = 3 * 2240
- H = 6720 meters
14. Therefore, the height of the tower is 6720 meters.

Conclusion:
The height of the tower is 6720 meters.