Math(Revised)

Phoebe and Tucker are 48m apart. Phoebe looks up at an angle of 43 degrees to see a hot air balloon hovering in the sky directly between her and Tucker. Tucker looks up at an angle of 31 degrees to see the hot air balloon.
A.) Explain how it could be determined who is closer to the balloon without doing any calculations.
B.) Determine the distance between the balloon and the person nearest the balloon. (I think I have to use sine but I'm not sure how to do this.)

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  1. Make a sketch, with P for Phoebe and T for Tucker as a horizontal line, so that PT = 45 m
    Let B be the position of the balloon, so that angle BPT = 43° and angle BTP = 31°
    Let AB be the height of the balloon with A directly below B on PT

    sin 43° = AB/BP ---> BP = AB/sin43
    sin 31° = AB/BT ---> BT = AB/sin31
    now we know, or should know, that sin 43>sin31
    So AB divided by the larger number would produce the smaller answer.
    So BP < BT
    (notice I did not do any calculations)

    To actually find PB, let PA = x, the AT = 48-x
    tan 43 = AB/x
    AB = xtan43

    tan31 = AB/(48-x)
    AB = (48-x)tan31

    so xtan43 = 48-xtan31
    xtan43 + tan31 = 48
    x(tan43+tan31) = 48
    x = 48/(tan43+tan31)

    now cos 43 = x/BP
    BP = x/cos43
    = (48/(tan43 + tan31) )(1/cos43)
    only now do we do any calculation ....

    I get BP = 42.8 m

    check my work please

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