related rates?

Question

James is flying a kite 300 ft above ground. A wind gust carries the kite horizontally away from james at a rate of 25ft per second. How fast must james let the string out when the kite is 500ft from him?

Answer 1

Raj, hello from nearby Itasca...

 

Here's a process you can try for related rates problems:

 

1.  Draw a picture of what is going on.  In this case, you've got a kite 300 feet above the ground, but it's not directly above the person flying the kite.  So, there's some horizontal distance between the person and the kite, and then there's the string connecting the person to the kite.

 

2.  Label stuff you don't know.

 

3.  Figure out an equation that somehow connects all of the stuff you don't know in the diagram.  For this problem, doesn't it have something to do with the sides of a right triangle?

 

4.  If there's anything in the diagram that is constant, then you can plug it in at this point.  In this problem, the wind is blowing the kite, so it's moving horizontally...which means we have to let out some string, so the length of the string is changing, too.  However, as far as I can tell, the idea is that the kite stays 300 feet above the ground at all times.  So, if you didn't already plug that into your equation, you can.  But you can't say that the kite is 500 feet from the person yet!  That's actually changing.  We'll deal with that part of the problem later, but you can't plug in values unless they are constant, and the 500 feet isn't constant.

 

5.  Now, you should have an equation with two variables in it.  Ideally, that's what happens.  In some problems, though, you actually end up with three variables at this point, which is yucky.  Sometimes, it's unavoidable, but in those cases, you can try to see if there's some other relationship between the variables so you can get rid of one of them.  (Maybe you know that y is twice x, for example, so you can substitute.)  In this problem, though, I think you should be okay.

 

6.  It's a related rate problem, so we need rates.  Rates are derivatives with respect to time (how much x changes as t changes?), so your next step is to take the derivative of your entire equation with respect to time.  Hint #1:  the derivative of a constant is still zero.  Hint #2:  the derivative of a variable has to involve the chain rule somehow.  The derivative of x, with respect to x, is just 1, but the derivative of x, with respect to time, isn't 1.  It's dx/dt.  This step is all about implicit differentiation, so if you're not clear on how to do that, let us know.

 

7.  You should now have an equation with variables and their derivatives in them.  Based on the situation given in the problem, now you get to plug in everything and then find the missing quantity.  Sometimes, you need to figure out something indirectly.  For example, in this problem, it tells you that the kite is 500 feet from him, but it doesn't explicitly tell you the horizontal distance at that same moment....but you should be able to figure that out from the diagram and your original equation.  The problem even gives you a rate, which you can plug in for one of the derivatives...because a derivative is a rate, right?

 

8.  You're down to an equation with one unknown quantity, which you should be able to solve for with some algebra.

 

Hope this helps, but let us know what questions you have!

 

Answer 2

The instantaneous velocity is 25 feet per second.  I thought it was going to be a clearly written calculus problem until I read it carefully. Q:?)

The horizontal velocity of the wind must be decreasing as the kite gets further from him.

I wonder if there are some subsequent questions dealing with acceleration or something like that.