Trig

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A billboard rests on top of a base. From a point level with the bottom of the base, the angle of elevation of the bottom of the billboard is 26 degrees and the top of the billboard is 31 degrees. From a point 21 meters further away, the angle of elevation to the top of the billboard is 17 degrees. Find the height of the billboard.

  • Trig -

    Let's make a diagram.
    Call the top of the billboard A and the bottom of the billboard E, and the point below AE we will call C, where C is at ground level.
    Label D as the first point of observation, and B as the second point of observation, so that BD = 21
    Let's look at triangle ABD.
    Angle BDA = 180-31 = 149º
    then angle BAD = 14º
    and angle B = 17º

    So by the sine law,
    AD/sin17 = 21/sin14
    We can find AD.
    Now let's look at triangle ADC which is right-angled.
    cos 31 = DC/AD, thus we can find DC
    also sin 31 = AC/AD, giving us AC
    and finally in triangle DEC
    tan 26 = EC/DC giving us EC

    so the billboard's height is AC - EC

    Do not round off preliminary answers, good luck.

  • Trig -

    Let a be the original horizontal distance from which the billboard is viewed.
    Let h be the height of the billboard.
    Let b be the height of the base.
    b/a = tan 26 = 0.4877
    (b+h)/a = tan 31 = 0.6009
    (b+h)/(a + 21) = tan 17 = 0.3057

    You have to solve three equations in three unknowns (a, b, h) to get h.

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