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Thread: Flagpole and Building

  1. #1
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    Flagpole and Building

    A 50-ft flagpole stands on the top of a 25-ft building. How far from the base of the building should a person stand if the flagpole and the building are to subtend equal angles at her eye, which is 5 ft above the ground?
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  2. #2
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    Re: Flagpole and Building

    Did you make a sketch?

    $\dfrac{20}{x} = \tan{\theta}$

    $\dfrac{70}{x} = \tan(2\theta)$

    solve the system for $x$ ...
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  3. #3
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    Re: Flagpole and Building

    when I solve the system, I get tan x = square root of 3/7

    x = 20 / square root of 3/7 = 30

    but the book answer is 40

    Flagpole and Building-flagpole.jpg
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    Re: Flagpole and Building

    My mistake ... I misinterpreted the set up. Let me work on it again.
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    Re: Flagpole and Building

    $\tan{\alpha} = \dfrac{5}{x}$

    $\tan{\beta} = \dfrac{20}{x}$

    $\tan(\theta+\beta) = \dfrac{70}{x}$

    $\alpha + \beta = \theta$

    a bit more complex of a system, but solvable ...
    Attached Thumbnails Attached Thumbnails Flagpole and Building-flagpole_prob.jpg  
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    Re: Flagpole and Building

    wOW. Very difficult to solve it algebraically
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    Re: Flagpole and Building

    There is a simpler solution possible using the angle bisector theorem ...

    The segment from eyeball to top of building cuts the opposite side in a ratio of 25:50, or 1:2

    Adjacent sides of large triangle (eyeball to base of building & eyeball to top of flag) will also be in a 1:2 ratio.

    Pythagoras ...

    $x^2+5^2=a^2$

    $x^2+70^2=(2a)^2 \implies x^2+70^2=4(x^2+5^2) \implies 4800=3x^2 \implies x = 40$
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    Re: Flagpole and Building

    wOW, it is very easy to solve using the theorem.
    Does the angle bisector always give adjacent sides the ratio 1:2?
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    Re: Flagpole and Building

    Quote Originally Posted by joshuaa View Post
    wOW, it is very easy to solve using the theorem.
    Does the angle bisector always give adjacent sides the ratio 1:2?
    see link ... click on the question mark (?)

    angle-bisector-theorem
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    Re: Flagpole and Building

    wOW, that's fantastic and very helpful. thanks a Lot.
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