Answer: the radius of the circular path is approximately 1637.58 meters.
Step-by-step explanation:
The centripetal force acting on the airplane is provided by the component of the gravitational force that acts towards the center of the circular path. This component is given by:
F_c = m * g * tan(banking angle)
Where:
F_c is the centripetal force
m is the mass of the airplane
g is the acceleration due to gravity
tan(banking angle) is the tangent of the banking angle
Now, the centripetal force is also given by the formula:
F_c = (m * v^2) / r
Where:
v is the speed of the airplane
r is the radius of the circular path
Equating the two expressions for F_c, we get:
(m * g * tan(banking angle)) = (m * v^2) / r
Canceling out the mass (m) on both sides of the equation, we have:
g * tan(banking angle) = v^2 / r
Solving for r, we get:
r = (v^2) / (g * tan(banking angle))
Substituting the given values:
v = 400 km/h = 400,000 m/h
g = 9.8 m/s^2
banking angle = 20°
Converting the speed to m/s:
v = 400,000 m/h * (1/3600) h/s = 111.11 m/s
Converting the banking angle to radians:
banking angle = 20° * (π/180) rad/° = 0.3491 rad
Now, substituting the values into the formula:
r = (111.11^2) / (9.8 * tan(0.3491))
r ≈ 1637.58 meters
Therefore, the radius of the circular path is approximately 1637.58 meters.