Given:
- Height of the building = 50 m
- Angle of depression from the top of the tower to the top of the building = 30 degrees
- Angle of depression from the top of the tower to the bottom of the building = 45 degrees

To find:
- Height of the tower
- Horizontal distance between the building and the tower

Assumptions:
- The ground is flat and level.

Approach:
1. Draw a diagram to represent the given information and visualize the situation.
2. Use trigonometric ratios such as tangent and sine to relate the angles of depression to the heights and distances.
3. Set up equations based on the trigonometric ratios and solve them to find the unknowns.

Solution:
Let's label the diagram as follows:

A: Top of the building
B: Bottom of the building
C: Top of the tower
D: Bottom of the tower
E: Position on the ground directly below the top of the tower

Step 1: Find the height of the tower
1.1. Consider the triangle ACD. The angle of depression at A is 30 degrees.
1.2. We can use the tangent function to relate the angle and the height of the tower:

tan(30) = height of the tower / horizontal distance CD

1.3. Since the height of the tower is unknown, let's represent it as 'x'.

tan(30) = x / CD

1.4. Rearranging the equation, we get:

x = CD * tan(30)

Step 2: Find the horizontal distance between the building and the tower
2.1. Consider the triangle ABE. The angle of depression at A is 30 degrees.
2.2. We can use the tangent function to relate the angle and the horizontal distance between the building and the tower:

tan(30) = height of the building / distance AE

2.3. Substituting the given value for the height of the building and rearranging the equation, we get:

AE = height of the building / tan(30)

2.4. Since the height of the building is 50 m and the tangent of 30 degrees is (√3) / 3, we can calculate the value of AE:

AE = 50 / (√3) / 3

2.5. Simplifying the expression, we get:

AE = 50 * 3 / √3

2.6. Rationalizing the denominator, we get:

AE = 50 * 3 * √3 / (√3 * √3)

AE = 50 * 3 * √3 / 3

AE = 50√3

Conclusion:
- The height of the tower is CD * tan(30).
- The horizontal distance between the building and the tower is 50√3.