# Air Traffic Controller and Related Rates along with Implicit Differentiation

• Salazar
In summary, to find the rate at which the distance between two planes is decreasing when they are converging at a right angle, we can use the equation dz/dt= 2x(dx/dt)+ 2y(dy/dt) and solve for dz/dt by setting x and y equal to the distances between the planes and using the given rates of dx/dt and dy/dt. To find the minimum distance between the planes, we can set dz/dt= 0 and solve for t, then use the position equations for x and y to find the distance at that time.
Salazar

## Homework Statement

"An air traffic controller spots two planes at the same altitude converging on a point as they fly at right angles to each other. One plane is 120 miles from the point and is moving at 400 miles per hour. The other plane is 160 miles from the point and has a speed of 450 miles per hour."
"a. At what rate is the distance between the planes decreasing?
b. If the controller does not intervene, how close will the planes come to each other?"

None given.

## The Attempt at a Solution

I first used x^2+y^2=z^2. I differentiated that and got 2x(dx/dt) +2y(dy/dt) = 2z(dz/dt).
The unknown being solved for was (dz/dt) for part and I had the rates for (dx/dt) and (dy/dt). I also had x and y, which were the distances.
I plugged it in and solved for (dz/dt). 2(160)(450) + 2(120)(400) = 2(200)(dz/dt) [z was found by using x^2+y^2=z^2].

For part b) I am a little lost on what to do. I am trying to find an extrema, the minimum for z, the distance between the planes. I am not sure how to set it up to find the minimum or what to do exactly. Can anyone please help? :\

It's pretty straight forward: yes, you want to minimize z where $z^2= x^2+ y^2$ and you do that by setting dz/dt= 0 and solving for t.

Of course, dz/dt= 2x(dx/dt)+ 2y(dy/dt) and you know that dx/dt= -400 and dy/dt= -420 (notice the signs- the two airplanes are moving toward the intersection of their routes so the distance between each and that intersection is decreasing. Of course, with x and y being the distance from each plane to that intersection, taking t= 0 when x is 120 and y is 160, x= 120- 400t and y= 160- 420t.

Your equation is dz/dt= 2(120-400t)(-400)+ 2(160-420t)(-420)= 0. Solve that for t, determine the position of each airplane at that t and find the distance between them.

HallsofIvy said:
It's pretty straight forward: yes, you want to minimize z where $z^2= x^2+ y^2$ and you do that by setting dz/dt= 0 and solving for t.

Of course, dz/dt= 2x(dx/dt)+ 2y(dy/dt) and you know that dx/dt= -400 and dy/dt= -420 (notice the signs- the two airplanes are moving toward the intersection of their routes so the distance between each and that intersection is decreasing. Of course, with x and y being the distance from each plane to that intersection, taking t= 0 when x is 120 and y is 160, x= 120- 400t and y= 160- 420t.

Your equation is dz/dt= 2(120-400t)(-400)+ 2(160-420t)(-420)= 0. Solve that for t, determine the position of each airplane at that t and find the distance between them.

Thanks, this really helped out. But just to be sure, it would be dz/dt= 2(120-400t)(-400)+ 2(160-450t)(-450)= 0, not using -420.

## 1. What is an air traffic controller and what does their job entail?

An air traffic controller is a person responsible for managing and monitoring air traffic in a designated airspace. Their job involves communicating with pilots, providing them with instructions and guidance to ensure safe and efficient travel, and monitoring weather and other factors that may affect flight operations.

## 2. What are related rates and how are they used in air traffic control?

Related rates are used in air traffic control to track the speed and direction of aircraft in relation to each other. By using mathematical equations and data from radar and other instruments, air traffic controllers can determine the rate of change of an aircraft's position and adjust their flight paths accordingly to avoid collisions.

## 3. What is implicit differentiation and why is it important for air traffic control?

Implicit differentiation is a method used to find the rate of change of a function that is not explicitly defined. In air traffic control, implicit differentiation is used to calculate the rate of change of aircraft positions and velocities, which is crucial for maintaining safe distances between planes and navigating them through crowded airspace.

## 4. How does air traffic control use technology to aid in their job?

Air traffic control uses various technologies such as radar, satellite systems, and computer systems to track and monitor air traffic. These technologies provide real-time data on the location, speed, and direction of aircraft, allowing controllers to make informed decisions and quickly respond to changes in flight paths.

## 5. What are some challenges that air traffic controllers face in their job?

Air traffic controllers face several challenges, including managing high volumes of air traffic, dealing with adverse weather conditions, and making split-second decisions in emergency situations. They also need to be able to communicate effectively with pilots from different countries and backgrounds and stay updated on the latest technology and procedures in the aviation industry.

• Calculus and Beyond Homework Help
Replies
2
Views
926
• Calculus and Beyond Homework Help
Replies
3
Views
481
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
470
• Calculus and Beyond Homework Help
Replies
2
Views
2K
• Calculus and Beyond Homework Help
Replies
12
Views
2K
• Calculus and Beyond Homework Help
Replies
6
Views
2K
• Calculus and Beyond Homework Help
Replies
40
Views
3K
• Calculus and Beyond Homework Help
Replies
6
Views
3K
• Calculus and Beyond Homework Help
Replies
5
Views
1K