An airbus A320 aircraft is cruising at an altitude of 10000 m. The aircraft is flying in a straight line away from Rachel, who is standing on the ground. If she sees the angle of elevation of the aircraft change from 70 degrees to 33 degrees in one minute, what is its cruising speed, to the nearest kilometer per hour?
The distance the airplane covers in that time is 10 km * (cot 33 - cot 70). Divide that by the time (1/60 h)
D = 15.3986 - 3.6397 = 11.759 km
V = ____ km/h
To find the cruising speed of the aircraft, we need to use trigonometry and the information given. We can use the concept of tangent to solve this problem.
Let's start by drawing a diagram to visualize the situation. We have Rachel standing on the ground and the aircraft flying in a straight line away from her. The angle of elevation of the aircraft changes from 70 degrees to 33 degrees in one minute.
/
/
/
/ <-- aircraft (changes from 70 to 33 degrees)
/
Rachel (on the ground)
Now, let's focus on a right triangle formed by Rachel, the aircraft, and a point directly beneath the aircraft. The altitude of the aircraft is given as 10,000 m, and we have two angles: 70 degrees and 33 degrees.
Using the tangent function, we can relate the angle to the side lengths of the triangle:
tangent(angle) = opposite / adjacent
Let's denote the distance traveled by the aircraft in one minute as "d" and the distance between Rachel and the point directly beneath the aircraft as "x".
For the initial angle of 70 degrees:
tan(70) = 10,000 / x
For the final angle of 33 degrees:
tan(33) = 10,000 / (x + d)
Now, we can solve these two equations simultaneously to find the value of "d".
tan(70) = 10,000 / x
tan(33) = 10,000 / (x + d)
Rearranging the first equation:
x = 10,000 / tan(70)
Substituting x into the second equation:
tan(33) = 10,000 / (10,000 / tan(70) + d)
Simplifying further:
tan(33) = tan(70) / (1 + d / tan(70))
To solve for "d", we can rearrange the equation:
d = (tan(33) / tan(70)) * (10,000 - 10,000 / tan(70))
Now, we can calculate the value of "d":
d ≈ (tan(33) / tan(70)) * (10,000 - 10,000 / tan(70))
Using a scientific calculator, we find that "d" is approximately 7,189.62 meters.
Since the aircraft travels this distance in one minute, to find the speed in meters per minute, we divide "d" by 1:
speed = 7,189.62 meters per minute.
Finally, we can convert the speed to kilometers per hour. There are 60 minutes in an hour and 1,000 meters in a kilometer:
speed = (7,189.62 * 60) / 1,000 ≈ 431.38 kilometers per hour.
Therefore, the cruising speed of the aircraft, to the nearest kilometer per hour, is approximately 431 km/h.