First of all, we need to covert the given angle from degrees to radians as calculations in trigonometry require the usage of radian measurement. The conversion formula is:
radians = degrees * pi/ 180
So, from the given data, the breeze's angle in radians would be:
breeze_rad = 11.48 degrees * pi/180 ≈ 0.20036379812894903 radians
This conversion is necessary because trigonometric functions in mathematics use radian measures.
Now, let's determine the x-direction component of the breeze's velocity. Since the breeze's velocity is given in an angled direction, it has both x and y components. Since we're only interested in the x-direction (horizontal movement), we calculate the component of breeze's velocity in x-direction using cos function:
breeze_x = breeze_velocity * cos(breeze_rad)
= 2.67 m/s * cos(0.20036379812894903 rad)
≈ 2.616584614009014 m/s
Finally, we can calculate the total velocity of the airplane relative to the ground, which is the vector sum of the airplane's own velocity and the x-component of the breeze's velocity:
total_velocity = plane_velocity + breeze_x
= 9.08 m/s + 2.616584614009014 m/s
≈ 11.696584614009014 m/s
Therefore, the magnitude of the plane's velocity relative to the ground, given the effect of the breeze is approximately 11.70 m/s.