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Thread: 3D question

  1. #1
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    3D question

    A pole, base at O, is on the west bank of a river. Two points, A and B, stand on the east bank of the river whose banks run parallel north to south. A to the south of O and B to the north. the distance AB is [tex]2a\sqrt{7}[/math and the angle AOB is $\displaystyle 150^o$. If the angle of elevation of the top of the pole, P, from A is $\displaystyle 45^o$ and $\displaystyle 30^o$ from B, find in terms of a the height of the pole and the width of the river.

    I know OP=OA and OA=$\displaystyle 4a\sqrt{7}\sin B$
    and $\displaystyle (A+B)=30^o$ but i don't know how to eliminate the sinB
    thanks
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  2. #2
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    Quote Originally Posted by arze View Post

    I know OP=OA and OA=$\displaystyle 4a\sqrt{7}\sin B$
    and $\displaystyle (A+B)=30^o$ but i don't know how to eliminate the sinB
    thanks
    OB = $\displaystyle 4a\sqrt{7}\sin A$ = sqrt{3}OP
    OA=$\displaystyle 4a\sqrt{7}\sin B$ = OP
    Sin(A)/sin(B) = sqrt{3}. But A = (30 - B)
    So sin(30 - B)/sin(B) = sqrt{3}. expand sin(30-B)
    (sin30*cos(B) - cos30*sinB)/sin(B) = sqrt{3}
    1/2*cot(B) - sqrt{3}/2 = sqrt{3}
    Solve the equation and find the angle B.
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