Given information:
- Angle of elevation of the top of the tower from a point on the same level as the foot of the tower is 30 degrees.
- On advancing 150 m towards the foot of the tower, the angle of elevation becomes 60 degrees.
- The height of the tower is 129.9 m.
- √3 = 1.732

To find:
The original distance between the point and the foot of the tower.

Solution:

Step 1: Drawing a diagram
Let's start by drawing a diagram to visualize the problem. We have a tower with its top and bottom marked, and a point on the same level as the foot of the tower. The angle of elevation from this point to the top of the tower is 30 degrees. On advancing 150 m towards the foot of the tower, the angle of elevation becomes 60 degrees.

Step 2: Setting up variables
Let 'x' be the original distance between the point and the foot of the tower. We are given that on advancing 150 m towards the foot of the tower, the angle of elevation becomes 60 degrees.

Step 3: Applying trigonometry
Using trigonometry, we can determine the height of the tower in terms of 'x' and solve for 'x'.

In a right triangle, the tangent of an angle is equal to the ratio of the opposite side to the adjacent side.

Using the first angle of elevation (30 degrees), we have:
tan(30) = height of the tower / x

Using the second angle of elevation (60 degrees), we have:
tan(60) = height of the tower / (x + 150)

Step 4: Solving the equations
We can rewrite the equations as follows:
1.732 = height of the tower / x
√3 = height of the tower / (x + 150)

Multiplying both equations by 'x' and (x + 150) respectively, we get:
1.732x = height of the tower
√3(x + 150) = height of the tower

Since both equations represent the height of the tower, we can equate them:
1.732x = √3(x + 150)

Step 5: Solving for 'x'
Now, we can solve the equation to find the value of 'x':

1.732x = √3(x + 150)
1.732x = √3x + 150√3
1.732x - √3x = 150√3
x(1.732 - √3) = 150√3
x = 150√3 / (1.732 - √3)

Step 6: Simplifying the expression
To simplify the expression, we can rationalize the denominator:

x = (150√3 / (1.732 - √3)) * ((1.732 + √3) / (1.732 + √3))
x = (150√3 * (1.732 + √3)) / ((1.732)^2 - (√3)^2)
x = (150√3 * (1.732 + √