Questions About Airplane Flight Paths And Parametric Equatio

Questions about airplane flight paths, parametric equations, and calculus derivatives

In this question, positions are given with reference to a Cartesian coordinate system whose x- and y-axes point due East and due North, respectively. Distances are measured in kilometres. An aeroplane flies in a straight line from City A, at (200, -100), to City B, at (500, -700). (a) (i) Find the equation of the line of flight of the aeroplane. (ii) Find the direction of travel of the aeroplane, as a bearing, with the angle correct to one decimal place. (b) After landing at City B, the aeroplane flies in a straight line in the direction N57° W to City C, then returns to City A in a straight line in the direction N16° E. (i) Draw a triangle showing the three flights and calculate the three angles of the triangle in degrees to one decimal place. (ii) Find the distance flown from City C to City A, rounded to the nearest kilometre. (c) The flight from City A to City B is observed from a detection tower at (500, -400). (i) Find parametric equations for this line of flight, with the parameter t where t=0 at City A and t=1 at City B. (ii) Write an expression for the square of the distance between the tower and the point at parameter t, in terms of t, and simplify. (iii) Determine the minimum distance between the aeroplane's flight path and the detection tower, rounding to the nearest kilometre, using this expression and the method of completing the square.

Paper For Above instruction

The given problem involves analyzing the flight path of an airplane navigating between multiple cities using coordinate geometry, vectors, and calculus. The solution encompasses deriving equations of lines, calculating bearings, constructing reaction diagrams, and applying calculus techniques for optimization and distance calculations. This comprehensive approach highlights the importance of analytical geometry in real-world navigation and travel planning.

Analysis of the Flight Path

The initial step involves determining the line of flight from City A (200, -100) to City B (500, -700). The slope m of this line is computed as:

m = (y2 - y1) / (x2 - x1) = (-700 - (-100)) / (500 - 200) = (-600) / 300 = -2

Using point-slope form with City A’s coordinates:

y - (-100) = -2(x - 200)

which simplifies to:

y + 100 = -2x + 400

Therefore, the equation of the flight line is:

y = -2x + 300

To find the bearing of the airplane’s travel, we interpret the slope as a directional vector:

rise = Δy = -600; run = Δx = 300

Considering the aircraft moves from A to B, the direction vector is (300, -600). The angle θ with respect to the east (x-axis) is given by:

θ = arctangent of (|Δy| / |Δx|) = arctangent (600 / 300) = arctangent (2) ≈ 63.4°

Since the Δy is negative, the movement is towards the south; thus, the bearing is approximately South 26.6° West, or, in the conventional bearing notation, 180° + 63.4° ≈ 243.4°. However, standard bearings are measured clockwise from due North, so:

bearing = 180° + 63.4° = 243.4°, which corresponds to approximately S 63.4° W. For the exact navigation bearing, considering compass conventions, the bearing from North is approximately 180° + 63.4° = 243.4°.

Constructing the Triangle of Flights and Computing Angles

Next, the airplane flies from City B (500, -700) in the direction N57° W toward City C, then back in the direction N16° E to City A. To analyze this, we must determine the locations of City C and the respective vectors.

The direction N57° W from City B implies a heading 57° west of due North. The directional vector is:

Component form: (sin 57°, cos 57°), but since it's towards North 57° West, the vector points towards ( -cos 57°, sin 57°). Numerical computation yields:

cos 57° ≈ 0.5446; sin 57° ≈ 0.8387

So, the vector from City B to City C is proportional to (-0.5446, 0.8387), scaled by the distance c to be determined.

Similarly, from City C to City A, in the direction N16° E, the vector is (cos 16°, sin 16°), with positive x and y components indicating a heading slightly east of due North.

Using these vectors and the coordinates, we can construct equations for the positions of City C, then apply the Law of Cosines to find the angles of the triangle formed by the three flights.

The distance from City B to City C (d1) and from City C to City A (d2) are unknowns, but by applying law of cosines with the known angles, the distances can be calculated accurately.

Calculating the Distance from City C to City A

Once the triangle is established, the Law of Cosines provides the distance from City C to City A:

d_{CA}^2 = d_{AB}^2 + d_{CB}^2 - 2 d_{AB} d_{CB} \cos \theta

where d_{AB} = distance from City A to City B, and θ is the angle at City C obtained from the triangle.

By plugging in the known distances and angles, the exact length from City C to City A is calculated, then rounded to the nearest kilometre.

Parametric Equations and Distance Minimization

Moving to the parametric analysis, for the flight from City A to City B, the parametric equations are derived as follows:

Let t ∈ [0, 1], then:

x(t) = x_A + t(x_B - x_A) = 200 + 300t

y(t) = y_A + t(y_B - y_A) = -100 - 600t

Similarly, the squared distance from the detection tower at (500, -400) to a point on the line is:

D^2(t) = (x(t) - 500)^2 + (y(t) + 400)^2

Substituting the parametric equations yields a quadratic in t, which is simplified and completed to find its minimum, indicating the closest approach of the plane to the tower. Applying derivatives and completing the square, the minimal distance is estimated and rounded to 1 km, ensuring practical relevance.

Conclusion and Implications

This analysis underscores the relevance of coordinate geometry, vector analysis, and calculus in modern navigation and flight planning. Calculating exact bearings, plotting flight paths, and optimizing distances based on real-world data are fundamental for safe and efficient aviation operations. The methods exemplified—line equations, parametric forms, vectors, the Law of Cosines, and calculus—are essential tools in aerospace navigation and engineering design, offering precise solutions vital for operational success and safety standards.

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