Calculation of the Angle
To calculate the angle through which a cyclist bends from the vertical when crossing a circular path, we need to consider the centripetal force acting on the cyclist.

Centripetal Force
The centripetal force is the force that keeps an object moving in a curved path and is directed towards the center of the circular path. In this case, the centripetal force is provided by the friction between the bicycle tires and the road.

Formula
The formula for centripetal force is given by:

F = m * a

Where:
F = Centripetal force
m = Mass of the cyclist
a = Centripetal acceleration

Centripetal Acceleration
The centripetal acceleration is the rate of change of velocity of the cyclist in the circular path. It is given by:

a = v^2 / r

Where:
v = Velocity of the cyclist
r = Radius of the circular path

Calculation of Velocity
To calculate the velocity of the cyclist, we can use the formula for linear velocity:

v = 2πr / t

Where:
π = Pi (approximately 3.14159)
r = Radius of the circular path
t = Time taken to complete one revolution

Given Values
Circumference of the circular path = 34.3 m
Time taken to complete one revolution = √22 seconds
Acceleration due to gravity (g) = 9.8 m/s²

Calculating Radius
The circumference of a circle is given by the formula:

C = 2πr

We can rearrange this formula to solve for the radius:

r = C / (2π)

Substituting the given values:

r = 34.3 / (2π)

Calculating Velocity
Now, we can calculate the velocity of the cyclist using the formula mentioned earlier:

v = 2πr / t

Substituting the values:

v = (2π * 34.3) / √22

Calculating Centripetal Acceleration
Using the formula for centripetal acceleration:

a = v^2 / r

Substituting the known values:

a = ((2π * 34.3) / √22)^2 / (34.3 / (2π))

Simplifying the equation:

a = (4π^2 * 34.3^2) / (√22 * 34.3)

Calculating Centripetal Force
Finally, we can calculate the centripetal force using the formula:

F = m * a

The mass of the cyclist is not given in the question, so we cannot calculate the exact centripetal force without that information.

Calculating the Angle
Unfortunately, without knowing the mass of the cyclist, we cannot directly calculate the angle through which the cyclist bends from the vertical. The angle depends on the centripetal force, which in turn depends on the mass of the cyclist.

However, we can make a rough estimate by assuming an average mass for a cyclist and then calculating the angle based on that.