Height of an object given angles of depression

AI Thread Summary
To determine the height of the hot-air balloon above the ground, two angles of depression (20° and 22°) to consecutive mileposts are used. The distance between the mileposts is one mile, creating two right triangles with the balloon's height as a common side. By applying trigonometric functions and the Pythagorean theorem, the height can be calculated. The discussion emphasizes the need for a clear setup of the right triangles to solve for the balloon's height. Ultimately, the problem is solvable with the provided information.
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Homework Statement



A hot-air balloon is floating above a straight road. To estimate their height above the ground, the balloonists simultaneously measure the angle of depression to two consecutive mileposts on the road on the same side of the balloon. The angles of depression are found to be 20o and 22o. How high is the balloon?


Homework Equations




  • the trigonometric functions

  • the Pythagorean theorem

The Attempt at a Solution



I have just tried constructing different right triangles, but always end up not having enough information to calculate side lengths and angles.
 
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From the balloon draw two angles of depression which meet ground at P and Q. Let OB be the height of the balloon from the ground. In the problem it is given that PQ = 1 mile. Let OP be x. Now you have two right triangles, OPB and OQB.
 
Is there enough given information to find a numerical value for OB?
 
There is enough information. The attachment should help once it's approved
 

Attachments

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
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