I was assisting my daughter with some homework; and on entering the answers online (they use some form of online homework app) it said that her answer was incorrect. I’m really not sure how my answer was marked as incorrect (at all), so I thought I’d post the same problem here and see what answers you all can come up with.
A plane is flying 2000 feet above sea level toward a mountain. The pilot observes the top of the mountain to be 17° above the horizontal, and immediately flies the plane at an angle of 20° above horizontal. The airspeed of the plane is 100 mph. After 5 minutes, the plane is directly above the top of the mountain. How high is the plane above the top of the mountain when it flies over? What is the height of the mountain? Round answers to the nearest 10 feet.
The hypotenuse is the path that the plane takes, with an airspeed of 100mph, that comes out to 44,000 feet of travel over the course of 5 minutes. That means that the line above the horizontal plane would be sin(20) * 44000, or 15,048 feet, and the horizontal distance travelled is sin(70) * 44,000, or 41,346 feet.
This means that the top of the mountain should be (sin(17) * 41,346)/sin(73), or 12,640 feet above horizontal, and 12,640 + 2000, or 14,640 high.
The airplane should be 15,048 - 12,640, or 2410 feet above the mountain top.
I suspect the question was written by a math teacher, not a pilot.
Given that there is no mention of wind speed in the question, it should probably have simply been written as “speed.” I would not be surprised if the question’s writer decided that “airspeed” would be more appropriate, as the question is about a plane’s travel, but it does open a can of worms for those who know that “airspeed” and “ground speed” can be different things.
I got that the plane is 2,410 ft above the mountaintop, and the mountain is 12,640 feet tall. Sounds like that what you got.
Apart from rounding errors in the calculations, I believe that’s correct. (And, since we’re rounding answers to the nearest 10 feet, I don’t think that’s a problem.) So maybe the key is entered incorrectly (or including / not including commas, etc., messes it up?)
Are you really confused by it, or do you just mean that it could’ve been written better? Are you really not sure what the intention of the question is?
I really was confused, as perhaps my lay understanding of air speed was somehow incorrect. It seemed to be a deliberate use of the term, and it didn’t make sense to me.
You are correct, but in this case I think we are in “spherical cow” territory, and making assumptions that the air is completely still, so your airspeed is an indication of just how far you move in a straight line along the flight path; to make the starting number the hypotenuse of the triangle, rather than the horizontal leg.
The use of the term here was obviously to indicate that this was the speed measured along the climbing trajectory (the hypotenuse), rather than the horizontal component.
The writer of the question presumably did not have any knowledge of aviation. The question really should have specified “still air”. It’s almost a disadvantage to be familiar with aviation here - you just have to look at the context of a high school trig question and make the assumption.
For what it’s worth, I tried solving the problem and got the same answers as you did here. Some of the details of my method were different (for instance, I worked in miles and only converted to feet at the end), but I think what I did was substantially the same.
As this problem has been solved I will use trigonometry to go off on a tangent and just state that it was lucky that the plane in the question was not a Germanwings plane or it might never have cleared the mountain. ducks and runs away
