Problem 47 - Algebra 2

Question

Air Traffic Control Suppose you are an air traffic controller directing the pilot of a plane on a hyperbolic fight path. You and anther air traffic controller from a different airport send radio signals to the pilot simultaneously. The two airports are 48 \(\mathrm{km}\) apart. The pilot's instrument panel tells him that the signal from your airport always arrives 100\(\mu \mathrm{s}\) (microseconds) before the signal from the other airport. a. To which airport is the plane closer? b. If the signals travel at a rate of 300 \(\mathrm{m} / \mu \mathrm{s}\) , what is the difference in distances from the plane to the two airports? c. Write the equation of the flight path. (Hint: \(k=2 a )\) d. Draw the hyperbola. Which branch represents the flight path?

Step-by-Step Solution

Answer

a) The plane is closer to the airport from whose signal arrives first. b) The difference in distances is 30 km. c) The equation of the flight path (hyperbola) is \(x^2/39^2 - y^2/9^2 = 1\). d) The branch of the drawn hyperbola that lies closer to the signal's point of origin (which is the first airport) represents the flight path.

Step 1: Identify Closer Airport

By definition, the signal that arrives first must have travelled a shorter distance than the one arriving later. Therefore, the airplane is closer to the airport from which it receives the signal 100μs earlier.

Step 2: Calculate Difference in Distances

Given that signals travel at a rate of 300 m/μs, it means that in the 100μs difference time, the later signal has travelled an additional \(300 m/μs * 100μs = 30000 m = 30 km\). So, a difference of 30 km exists between the distances from the plane to the two airports.

Step 3: Writing Equation of Flight Path

It is given that the two airports are 48 km apart and the plane is at a different distance to each airport (with a difference of 30 km between these distances). If we consider a coordinate system with the plane at the origin and two airports on the x-axis, then we need two points on our x-axis corresponding to the two airports. The midpoint between these two airports will then be 24 km from each airport. One airport is \(24 km - 15 km = 9 km\) from the plane, the other is \(24 km + 15 km = 39 km\). The equation of the hyperbola is then \(x^2/39^2 - y^2/9^2 = 1\).

Step 4: Draw Hyperbola and Identify the Flight Path

Draw two vertical lines (the asymptotes) which are 9 km and 39 km away from the plane (the center). Sketch the hyperbola that fits into this guideline. It will have branches that open to the right and left. The flight path is represented by the branch that is closer to the airport from which the signal arrives first or, mathematically, on the side of the x-axis where the airport's x-coordinate lies.

Key Concepts

Air Traffic Control Mathematics

Air traffic control relies heavily on mathematics to ensure the safety and efficiency of aircraft navigation. One of the mathematical concepts used is the hyperbolic flight path. In air traffic control, hyperbolas are used to determine the plane's position relative to two radio signals from different airports. Given two fixed points (airports), the time difference of radio signals received helps establish which airport is closer. This system is crucial for air traffic controllers to effectively manage aircraft paths, ensuring they avoid collisions and maintain efficient travel routes. Mathematics simplifes these processes, making air traffic control smoother and more precise.

Distance Calculation

Calculating the distance between objects is a fundamental aspect of navigational mathematics, particularly in air traffic control. In the given exercise, we determine the difference in distances based on the speed of signals and the time delay between their arrivals.

• Signals travel at 300 meters per microsecond.
• Time difference provided is 100 microseconds.

Thus, the difference in distance is calculated as: \(300 \, \text{m/μs} \times 100 \, \text{μs} = 30000 \, \text{m} = 30 \, \text{km}.\)Thus, the airplane is 30 km further from one airport than the other. Understanding these calculations helps estimate the plane's position more accurately, which is essential for ensuring safe travel paths.

Coordinate Geometry

Coordinate geometry helps in visualizing and solving problems involving shapes and distances in a plane. It involves using coordinates to derive an equation representing a particular figure, such as a hyperbola. In the context of air traffic control, we place the plane at the origin and position two airports along the x-axis.

• The total distance between airports is given as 48 km.
• Midpoint along the x-axis helps distribute distances equally, at 24 km from each airport.
• Using the distances and midpoint, the hyperbola's equation is formed.

For this situation, the equation of the hyperbola becomes \(\frac{x^2}{39^2} - \frac{y^2}{9^2} = 1.\)This equation provides a mathematical representation of the aircraft's path with respect to the airports.

Radio Signal Speed

Radio signals play a vital role in navigation and positioning techniques. These signals travel at a constant speed, which is crucial for calculating distances. In air traffic control scenarios, signals travel at the speed of light, around 300 meters per microsecond.

• This constant speed allows the determination of time and distance differences between two points.
• The speed being uniform helps simplify calculations.

With the constant speed of radio signals, we can accurately determine how distances correlate with the time it takes signals to travel. This principle assists air traffic controllers in continuously monitoring aircraft positions, enhancing navigation accuracy.