# Rates of change question

Problem:
A plane flying horizontally at an altitude of 1 mile and a speed of 500 mph passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 miles away from the station.

d(distance)/dt = sqrt(3)*250

I know this is the correct answer: looked in back of book.

Questions:

1. What exactly is d(distance)/dt? is it the velocity that the plane must go along the hypotenuse inorder to be at the same place at the same time if it where traveling along the horizontal?

2. Is this d(distance)/dt only constant at this moment in time? and not say 2,3,4.5... hours from its starting position?

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Mark44
Mentor
Problem:
A plane flying horizontally at an altitude of 1 mile and a speed of 500 mph passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 miles away from the station.

d(distance)/dt = sqrt(3)*250

I know this is the correct answer: looked in back of book.

Questions:

1. What exactly is d(distance)/dt? is it the velocity that the plane must go along the hypotenuse inorder to be at the same place at the same time if it where traveling along the horizontal?
It's the instantaneous rate of change of distance. It's not really the plane's velocity, since that would be a measure of its speed through the air (its airspeed, as measured by a pitot tube - not sure that they still use those) or its ground speed (its horizontal speed). The derivative here in your problem is measuring the rate at which the distance from a point on the ground is changing.
2. Is this d(distance)/dt only constant at this moment in time? and not say 2,3,4.5... hours from its starting position?
It's not constant at all. The number you gave as the answer is the value of this derivative at a particular moment. Before that time, the value would be less, and after that time, the value would be greater.

Is there anyway to find d(theta)/dt ?

Mark44
Mentor
Sure, if you can use trig to find an equation that relates the distance from the radar station to the angle theta, then you can differentiate to find another equation that represents the derivatives of the variables in the first equation.

That's pretty much what "related rates" questions are all about.

blB
I think finding dtheta/dt would be a roundabout way of doing this problem. You need to draw a triangle with a relation to the altitude, the distance between the plane and the radar station on the ground, and the speed of the plane related to time ( if the plane is going 500m/h then every hour it goes 500miles so 500t would give you a distance.

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I think finding dtheta/dt would be a roundabout way of doing this problem. You need to draw a triangle with a relation to the altitude, the distance between the plan and the radar station on the ground, and the speed of the plan related to time ( if the plan is going 500m/h then every hour it goes 500miles so 500t would give you a distance.
You are absolutly right, it would be much longer to find the change in the angle...
I was just curious to know if it was possible because I believe it will help in problem solving if I know other ways on how to solve one problem.. and that is why I asked.