What is the rate of approach to the tower?

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In summary, the conversation is about finding the rate of approach of an airplane flying near a radar tower. The airplane is initially 3 miles west and 4 miles high, with a ground speed of 450 mph and climbing at a rate of 5 mph. The position function is given as [-3, 0, 4] and the velocity vector is [450, 0, 5]. The rate of approach is found by taking the derivative of the position function and multiplying it by the velocity vector. The answer is 266 mph.
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bodensee9
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Homework Statement


Hello:
An airplane is flying near a radar tower. At the instant it is exactly 3 miles due west of the tower. It is 4 miles high and flying with a ground speed of 450 mph and climbing at rate of 5 mph. If at the instant it is flying east, what is the rate of approach to the tower.

So I think this means that its position is [-3 0 4] if we take the tower to be located at [0 0 0] and its velocity is [450 0 5]. I thought that the derivative of the position function evaluated at [-3 0 4] and multiplied by the velocity vector will give us the rate of approach. But would the position function be [450t 0 5t]? This looks wrong.

Thanks!
 
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  • #2
If you put t=0 into the position function you should get [-3,0,4], right? Doesn't that make the position function s(t)=[-3,0,4]+t*[450,0,5]?
 
  • #3
Hello:

Yes, but for some reason, I'm supposed to find the rate of approach (the tower). And the answer is 266 mph. But I'm not sure how I'm supposed to get that answer from what I have?

Thanks.
 
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  • #4
bodensee9 said:
Hello:

Yes, but for some reason, I'm supposed to find the rate of approach (the tower). And the answer is 266 mph. But I'm not sure how I'm supposed to get that answer from what I have?

Thanks.

You need to find d/dt(|s(t)|). What's |s(t)|?
 

1. What are directional derivatives?

Directional derivatives are a mathematical concept used to measure the rate of change of a function in a specific direction. They are commonly used in multivariate calculus to determine the slope of a function in a particular direction.

2. How do you calculate directional derivatives?

The directional derivative of a function can be calculated by taking the dot product of the gradient of the function and a unit vector in the desired direction. This can be represented by the formula Df(x,y) = ∇f(x,y) · u, where ∇f(x,y) is the gradient and u is the unit vector.

3. What is the relationship between directional derivatives and partial derivatives?

Directional derivatives are closely related to partial derivatives, as they are both used to measure the rate of change of a function. However, directional derivatives measure the rate of change in a specific direction, while partial derivatives measure the rate of change in a single variable while holding all other variables constant.

4. What is the significance of directional derivatives in real-world applications?

Directional derivatives are used in various fields such as physics, engineering, and economics to analyze and optimize functions. They can be used to determine the direction of steepest ascent or descent for a function, which is useful in optimization problems.

5. Can the directional derivative be negative?

Yes, the directional derivative can be negative. It represents the rate of change of a function in a specific direction, so if the function is decreasing in that direction, the directional derivative will be negative. Similarly, if the function is increasing, the directional derivative will be positive.

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