Answer:
The plane travels at a speed of approximately 262.768 miles per hour.
Step-by-step explanation:
The absolute velocity of the plane ([tex]\vec v_{P}[/tex]) is the vectorial sum of vectors from engines ([tex]\vec v_{E}[/tex]) and wind flow ([tex]\vec v_{W}[/tex]). All vectors are measured in miles per hour. That is:
[tex]\vec v_{P} = \vec v_{E} + \vec v_{W}[/tex] (1)
Let suppose that both north and east semiaxes are positive.
If we know that [tex]\vec v_{E} = (0\,mph, 300\,mph)[/tex] and [tex]\vec v_{W} = (-40\,mph\cdot \sin 20^{\circ}, -40\,mph\cdot \cos 20^{\circ})[/tex], then the resultant velocity of the plane is:
[tex]\vec v_{P} = \vec v_{E} + \vec v_{W}[/tex]
[tex]\vec v_{P} = (0\,mph, 300\,mph) + (-40\,mph\cdot \sin 20^{\circ},-40\,mph\cdot \cos 20^{\circ})[/tex]
[tex]\vec v_{P} = (0\,mph - 40\,mph\cdot \sin 20^{\circ}, 300\,mph -40\,mph\cdot \cos 20^{\circ})[/tex]
[tex]\vec v_{P} = (-13.681\,mph, 262.412\,mph)[/tex]
The resultant speed of the plane is determined by Pythagorean Theorem:
[tex]\|\vec v_{P}\| = \sqrt{\left(-13.681\,mph\right)^{2}+\left(262.412\,mph \right)^{2}}[/tex]
[tex]\|\vec v_{P}\| \approx 262.768\,mph[/tex]
The plane travels at a speed of approximately 262.768 miles per hour.
