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I have a question for the basic trigonometry instructor (re: example for the angle of depression):  Why wouldn't you solve for the hypotenuse since the airplane isn't on the ground?  Serving, Johnny
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Johnny.Christian

by Johnny.Christian at April 23, 2010

Regarding the plane in the example for the angle of depression:  Contrary to the flagpole example given in the answer below, the plane isn't on the ground.  If I travel from Florida to the state of Washington, you aren't going to tell me the horizontal component of the distance, you will tell me the straightline distance.  Likewise, If I am on that plane, I don't want only enough fuel to get me that horizontal distance, because my plane is in the air and must travel the full distance of the hypotenuse or it will crash.  So my question remains, given the plane's actual flight pattern, why wouldn't you solve for the hypotenuse since the airplane isn't on the ground?  Thank you!  Serving In Christ, Johnny

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Some lively debate here. I think Johnny has a point. "How far away is [the airplane] from the airport?" could be interpreted to mean the distance along the hypotenuse.  Clearly, Brian wants us to find the horizontal distance between the plane and the airport.  However, both problems are interesting.  Rounding to the nearest integer foot, we get 421526 feet for the horizontal distance and 421783 feet for the length of the hypotenuse.  The difference is only 257 feet.  Isn't that cool?  If you are sweating out a problem like this during a test, I think it would be fair to ask your teacher to specify which distance he or she wants.  If that's not an option, give both answers!  The airplane is diagonally 421783 feet from the airport, but horizontally it is 421526 feet from the airport. 
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Norm Norm April 24, 2010

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Nope. thts how far the plane will travel. when measuring how far it is, we take the base not the hypotenuse, even if its not on the ground. its pretty much same as any ordinary question. for example how far is the pole from the man, we take the base as the answer, not the hypotenuse
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Sandesh Sandesh April 23, 2010

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I disagree with this answer above.  The base of the pole is on the same plane as the man; so to measure from the man to the base of the pole would be appropriate.  If you were to ask me how far the top of the flagpole is from the man, you certainly wouldn't tell me the horizontal distance.  My plane is like the top of the flagpole.  The question in the problem has a vertical component that cannot be ignored which makes the actual distance it will travel farther than the simple horizontal distance.   If this plane only has enough gas to get it the horizontal component of the distance (as if it were a bus), it would crash short of the airport, because it would run out of fuel because the plane doesn't fly the base, it flies the hypotenuse.  
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Johnny.Christian Johnny.Christian April 23, 2010

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i dont agree. i think you're getting confused with two of these termsi) how far is the plane from the airport?ii) how far should the plane travel?which aren't the same-and in this particular question, we should be calculating (i) not (ii).and you would have been correct if it were asking how far should the plane travel, in which we would find the hypotenuse.You can use the practical examples to help you here. When you are in a plane going somewhere, and it says you have 1000 miles left, its telling you the distance from the ground to the airport, not from the plane. It is also using the base, not the hypotenuse.
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Sandesh Sandesh April 24, 2010

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Okay, Sandesh, if this is simply a mathmatical "convention" to separate #1 from #2 in your answer, then we are stuck with it; but you won't convince me it makes sense, because that isn't how we teach "distance".  We teach distance as being the hypotenuse when there is an "x" value and a "y" value to the separation of two objects.  It is quite clear from this problem that you have an "x" value and a "y" value that separates the two and therefore it should be the hypotenuse that we consider the distance, if we are going to stay mathematically consistent.  If you are right, then the brain trust who has designed how we study mathematical concepts has made an inconsistent decision for whatever reason they chose to, and I just have to deal with that.  Just realize that people who actually think about these concepts are going to struggle with the apparent inconsistency in training that the hypotenuse is normally distance, unless you are dealing with something that is not on a graph.
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Johnny.Christian Johnny.Christian April 24, 2010

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