Use Your Answer To Part Cii And The Method Of Completing Th
Use Your Answer To Part Cii And The Method Of Completing Th
Use your answer to part (c)(ii), and the method of completing the square, to determine the distance, to the nearest kilometre, between the aeroplane and the aircraft detection tower at the point on the line of flight of the aeroplane where it is closest to the aircraft detection tower. (The distance required is the ‘horizontal’ distance; that is, the distance between the aircraft detection tower and the point on the ground immediately below the aeroplane.)
Paper For Above instruction
The problem involves calculating the shortest horizontal distance between an airborne aircraft and a fixed detection tower, given certain parameters of the aircraft's flight path and the mathematical model describing its position relative to the tower. Employing techniques such as completing the square will facilitate finding the relevant minimum value of a quadratic function that models the squared distance, allowing for a precise calculation of the minimal horizontal distance to the nearest kilometre.
Introduction
Accurately determining the closest point between an aircraft and a detection tower is crucial for navigation safety and radar system efficiency. When the aircraft’s position at any time can be expressed as a quadratic function of time or position, the shortest horizontal distance can be found by using calculus or algebraic methods such as completing the square. In this context, the problem centers on translating the flight data into a quadratic function that models the squared distance from the tower, then identifying its minimum point.
Mathematical Modeling of the Aircraft’s Flight Path
Suppose the aircraft's horizontal position relative to the tower is represented by a quadratic function, say \(d(x) = ax^2 + bx + c\), where \(x\) is a parameter such as time or horizontal displacement. Here, the coefficient \(a\) indicates the curvature of the parabola representing the flight path, while \(b\) and \(c\) encode initial position and velocity components. Accurate values for these coefficients derive from the given data or previous parts of the problem.
The squared distance \(D^2(x)\) from the tower at the point \(x\) can be expressed as:
\[
D^2(x) = (x)^2 + (d(x))^2,
\]
where \(x\) is the horizontal distance from the tower, and \(d(x)\) reflects the altitude or another relevant vertical displacement component.
Completing the Square to Find the Minimum
To determine the point where the aircraft is closest to the detection tower, we need to find the value of \(x\) that minimizes \(D^2(x)\). This involves differentiating \(D^2(x)\) with respect to \(x\) and setting the derivative to zero. Alternatively, algebraic methods like completing the square are employed for quadratic functions, especially when an explicit formula for the minimum is desired without calculus.
Suppose, after substituting the known parameters from the problem, \(D^2(x)\) simplifies to a quadratic form:
\[
D^2(x) = Ax^2 + Bx + C,
\]
where \(A > 0\) guarantees a minimum. Completing the square involves rewriting this as:
\[
D^2(x) = A\left(x^2 + \frac{B}{A}x\right) + C = A \left[\left(x + \frac{B}{2A}\right)^2 - \left(\frac{B}{2A}\right)^2\right] + C,
\]
which simplifies to:
\[
D^2(x) = A \left(x + \frac{B}{2A}\right)^2 + \left(C - \frac{B^2}{4A}\right).
\]
The minimum occurs at \(x = -\frac{B}{2A}\), providing the optimal point on the line of flight where the aircraft is closest to the tower.
Calculating the Minimum Distance
Once the value of \(x = -\frac{B}{2A}\) is found, substituting back into the original distance function yields the minimum squared distance. Taking the square root of this value gives the actual minimum distance. Rounding this to the nearest kilometre completes the task—specifically, calculating \(\sqrt{D^2(x)}\) at the point of minimum.
Practical Application and Numerical Example
For illustrative purposes, assume the quadratic function modeling the squared distance is:
\[
D^2(x) = 4x^2 - 8x + 13.
\]
Completing the square:
\[
D^2(x) = 4(x^2 - 2x) + 13 = 4\left[(x - 1)^2 - 1\right] + 13 = 4(x - 1)^2 + 9.
\]
The minimum occurs at \(x = 1\), where the squared distance is \(9\). The minimum linear distance is therefore \(\sqrt{9} = 3\) units. Rounding to the nearest kilometre yields a distance of 3 km.
Conclusion
Through algebraic manipulation and the method of completing the square, it is possible to identify the specific point on the flight path at which the aircraft is closest to the detection tower. Critical steps involve deriving the quadratic form of the squared distance function, completing the square to find its vertex, and calculating the minimum distance. This approach provides a clear, algebraic pathway to solving optimization problems in flight path analysis, emphasizing the interplay between geometry and algebra in real-world applications.
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