Note: Given the circular track has a length of 440m and the cyclist completes one full round in 20 seconds, we need to find the angle that his cycle makes with the vertical.

1. Calculate the circumference of the circular track:
The circumference of a circle is given by the formula:
C = 2πr, where C is the circumference and r is the radius of the circle.
Since we are given the length of the track, which is the circumference, we can calculate the radius using the formula:
r = C / (2π)

Given that the length of the track is 440m:
C = 440m
r = 440m / (2π)

2. Calculate the time taken to complete one full round:
Since the cyclist completes one full round in 20 seconds, the time taken to complete one full round is equal to the time period of the cycle.

Time taken to complete one full round = 20 seconds

3. Calculate the angular velocity:
Angular velocity is defined as the rate of change of angular displacement with respect to time. It is given by the formula:
Angular velocity (ω) = θ / t, where θ is the angular displacement and t is the time taken.

In one full round, the angular displacement is 2π radians (360 degrees). Therefore, we can calculate the angular velocity as follows:
Angular velocity (ω) = 2π / 20

4. Calculate the angle the cycle makes with the vertical:
The angle the cycle makes with the vertical can be calculated using the concept of angular displacement.

In one complete round, the cycle covers an angular displacement of 2π radians. Let's assume the angle the cycle makes with the vertical is θ.

We can set up a proportion to find θ:
2π radians corresponds to a full round (360 degrees)
θ radians corresponds to the angle the cycle makes with the vertical

Using the proportion, we can calculate θ as follows:
θ = (θ radians / 2π radians) * 360 degrees

Simplifying the above equation, we get:
θ = (360 / 2π) * θ radians

5. Substitute the values to find the angle:
Substituting the value of θ from the previous equation, we can find the angle the cycle makes with the vertical.

θ = (360 / 2π) * θ radians
θ = (360 / 2π) * (2π / 20)

Simplifying the above equation, we get:
θ = 18 degrees

Therefore, the angle that the cyclist's cycle makes with the vertical is 18 degrees.