stacey is standing 5 miles west of the control tower at an airport. A plane took off traveling directly north of the control tower at a rate of 180 mph along the ground at a constant angle of 15 degrees.
A)after 12 minutes, what is the angel between Stacey and the plane along the ground?
B)after 12 minutes, what is the angle of elevation between Stacey and the plane?
C) how far is the plane from Stacey?
180mph = 3 mi/min
using x,y,z coordinates,
Stacey is at (-5,0,0)
At time t minutes,
The plane is at (0,3t,3tan15° t) = (0,3t,0.8t)
So, at time t, the angle θ from Stacey to the point under the plane, is
tanθ = 3t/5
So at 12 minutes, tanθ = 36/5
the angle of elevation θ from Stacey to the plane is
tanθ = 9.6/5
the distance at time t is
d^2 = 5^2+(3t)^2+(.8t)^2
so at t=12,
d^2 = 25+1296+92.16
d = 37.6 miles
To solve this problem, we will use trigonometry. Let's break down the information given and solve each part of the problem step by step.
A) To find the angle between Stacey and the plane along the ground after 12 minutes, we need to determine the horizontal distance between Stacey and the plane at that time.
Since the plane is traveling directly north at a rate of 180 mph and 12 minutes is 1/5th of an hour (since there are 60 minutes in an hour), we can calculate the horizontal distance as follows:
Horizontal distance = Speed * Time = 180 mph * (12 minutes / 60) hours = 36 miles.
Now, we have a right-angled triangle formed by Stacey, the vertical distance, and the horizontal distance. We know that the vertical distance (opposite side) is 5 miles (since Stacey is standing 5 miles west of the control tower) and the horizontal distance (adjacent side) is 36 miles. To find the angle between them (θ), we can use the inverse tangent function:
θ = arctan(opposite/adjacent)
θ = arctan(5/36) ≈ 7.93°
So, after 12 minutes, the angle between Stacey and the plane along the ground is approximately 7.93 degrees.
B) To find the angle of elevation between Stacey and the plane after 12 minutes, we can use the given constant angle of 15 degrees.
The angle of elevation is the angle formed between the horizontal ground and the line of sight from Stacey to the plane. Since the plane is initially traveling at an angle of 15 degrees (with respect to the horizontal), this angle will remain constant throughout.
Therefore, after 12 minutes, the angle of elevation between Stacey and the plane is 15 degrees.
C) To find the distance between Stacey and the plane after 12 minutes, we can use the Pythagorean theorem.
In the right-angled triangle formed by Stacey, the horizontal distance, and the vertical distance, we can use the following formula:
Distance^2 = Horizontal distance^2 + Vertical distance^2
Distance^2 = 36^2 + 5^2
Distance^2 = 1296 + 25
Distance^2 = 1321
Distance = √1321 ≈ 36.34 miles
Therefore, the plane is approximately 36.34 miles away from Stacey after 12 minutes.