An airplane travels due west for 2.5 hours at 340 miles per hour. Then it changes course to S51°W. Find the airplanes distance from its point of departure and its bearing, after a total flight time of 4.5 hours.
Using the law of cosines, the distance z is
z^2 = (2.5*340)^2 + (2*340)^2 - 2(2.5*340)(2*340)cos(141°)
z = 1443
bearing: 90-θ where
tanθ = (2*340*cos51°)/(2.5*340+2*340*sin51°)
θ = E 17.25° N
or 72.75°
To find the distance from the airplane's point of departure and its bearing after a total flight time of 4.5 hours, we can break down the given information and solve step by step.
First, let's calculate the distance traveled during the initial westward flight.
Distance = Speed × Time
Distance = 340 miles/hour × 2.5 hours
Distance = 850 miles
After traveling west for 2.5 hours, the airplane changes its course to S51°W. We need to calculate the distance and bearing for this leg of the flight.
To find the distance, we can use trigonometry. We have the angle given as S51°W, but we need to convert it to the direction perpendicular to the angle, which is N39°W.
Now, we can calculate the distance using the cosine rule:
Distance = √(850² + d² - 2 × 850 × d × cos(39°))
Squaring both sides:
Distance² = 850² + d² - 2 × 850 × d × cos(39°)
Next, let's calculate the flight time for the second leg using the equation:
Time = Distance / Speed
4.5 hours - 2.5 hours (initial flight time) = 2 hours (time for the second leg)
Now, we know the time for the second leg is 2 hours. We can solve for d using the given distance formula:
Distance² = 850² + d² - 2 × 850 × d × cos(39°)
d² - 2 × 850 × d × cos(39°) - Distance² + 850² = 0
Now we need to find the positive root using the quadratic formula:
d = (-b + √(b² - 4ac)) / 2a
where a = 1, b = -2 × 850 × cos(39°), and c = 850² - Distance².
Once we find the value of d (the distance of the second leg), we can add it to the initial distance traveled (850 miles) to find the total distance from the airplane's point of departure.
Total distance = Initial distance + Distance of the second leg
Finally, to find the bearing after a total flight time of 4.5 hours, we can use trigonometry. We already know the angle of the second leg is N39°W.
Bearing = 180° + angle of the second leg
Now that we have all the necessary calculations and steps, you can plug in the values into the formulas and equations mentioned above to find the final answers.