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 'Finding the number of ways to arrange identical balls in a circle (3 different colors)'

I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
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Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
View full post »

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