Rates of change question

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In summary, the problem involves finding the rate at which the distance from a plane flying horizontally at 1 mile altitude and 500 mph speed to a radar station is increasing when the plane is 2 miles away from the station. The solution involves using the formula d(distance)/dt = sqrt(3)*250 and understanding that it represents the instantaneous rate of change of distance. It is not constant and the value given is only for a particular moment in time. Finding d(theta)/dt would be a roundabout way of solving the problem and it would be more efficient to approach it through trigonometry.
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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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  • #2
Miike012 said:
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.
Miike012 said:
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.
 
  • #3
Is there anyway to find d(theta)/dt ?
 
  • #4
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.
 
  • #5
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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  • #6
blB said:
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.
 

1. What is a rate of change?

A rate of change is a mathematical concept that measures the speed at which a variable changes over a specific period of time. It is usually represented as a slope or a derivative in calculus.

2. How do you calculate a rate of change?

To calculate a rate of change, you need to find the difference between the final and initial values of a variable, and divide that by the corresponding difference in time. This is represented as "change in y over change in x" or "∆y/∆x".

3. What is the unit of measurement for a rate of change?

The unit of measurement for a rate of change depends on the variables involved. For example, if the rate of change is measuring distance over time, the unit could be meters per second. It is important to include the unit in your answer to make it meaningful.

4. How is a rate of change represented graphically?

A rate of change can be represented graphically as the slope of a line on a graph. The steeper the slope, the higher the rate of change. Alternatively, in a curve, the rate of change can be represented as the instantaneous slope at a specific point, which is the derivative.

5. What are some real-world applications of rates of change?

Rates of change have many real-world applications, such as calculating the speed of a moving object, determining the growth rate of a population, or analyzing the efficiency of a system. It is also used in fields such as physics, economics, and engineering to make predictions and solve problems.

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