Problem Statement:
The problem states that there are two towers of heights x and y standing on level ground. The angles subtended by the tops of the towers at the center of the line joining their feet are 30 degrees and 60 degrees respectively. We need to find the ratio x:y.

Solution:
To solve this problem, we can make use of basic trigonometric concepts and the properties of similar triangles.

Step 1: Drawing the Diagram
To better understand the problem, let's draw a diagram. Draw two vertical lines to represent the towers and a horizontal line to represent the ground. Label the height of the first tower as x and the height of the second tower as y. Mark the center of the line joining the feet of the towers as point O. Label the points where the tops of the towers touch the horizontal line as A and B respectively.

Step 2: Analyzing the Given Information
From the problem statement, we know that the angles subtended by the tops of the towers at point O are 30 degrees and 60 degrees. Let's label these angles as θ and φ respectively.

Step 3: Applying Trigonometric Functions
We can use the trigonometric functions to relate the given angles and the heights of the towers.

Step 3.1: Tower with Angle of 30 degrees
Consider triangle OAB. The angle AOB is 30 degrees, and we know that the angle at O is 90 degrees. We can use the trigonometric function tangent (tan) to relate the angle and the height of the tower.

tan(30 degrees) = x / OA

Simplifying this equation gives us:

1 / √3 = x / OA

Step 3.2: Tower with Angle of 60 degrees
Consider triangle OAB. The angle AOB is 60 degrees, and we know that the angle at O is 90 degrees. We can again use the trigonometric function tangent (tan) to relate the angle and the height of the tower.

tan(60 degrees) = y / OB

Simplifying this equation gives us:

√3 = y / OB

Step 4: Applying Similar Triangles
From the given information, we can observe that triangle OAB is an isosceles triangle. This means that the sides OA and OB are equal in length.

Since the triangles OAB and OBA are similar, we can write the following ratio:

OA / OB = OB / OA

Simplifying this equation gives us:

(OA)^2 = (OB)^2

Substituting the values from the previous equations gives us:

(x / (1 / √3))^2 = (y / √3)^2

Simplifying further:

(x^2 / (1 / 3)) = (y^2 / 3)

Cross-multiplying and simplifying:

3x^2 = y^2

Taking the square root of both sides:

√3x = y

Therefore, the ratio x:y is √3:1 or approximately 1.732:1.