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Math Help - adventitious angle driven problem

  1. #1
    Senior Member pacman's Avatar
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    adventitious angle driven problem

    Mario drives at the average speed at all times with his old car. He makes it from point A direct to point C in 30 minutes; from point A through point B to point C in 35 minutes; and from point A through point D to point C in 40 minutes. Distance from point B to point C is 10 kilometers, the same distance from point C to point D. By the way angle C is 90 degrees. Points A, B, C and D are 4 corners of a quadrilateral and labeled in a clockwise manner. How fast does Mario drive in km/hour?
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  2. #2
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    I'm assuming you're familiar with Law of Sines/Cosines.
    To write a FULL step by step solution here would be almost impossible;
    classroom environment required (blackboard, chalk.....).

    I'll give you a "how to get started".

    Draw an approximate representative diagram (quad ABCD); join AC.

    Let AB = b, AC = c, AD = d, average speed = x.

    Get b,c,d in terms of x (assuming you're familiar with speed = distance/time):
    b = (7x - 120) / 12, c = x / 2, d = (2x - 30) / 3
    (notice that x > 120/7)

    Now you need to calculate a few angles, like B; using triangle ABC:
    B = arccos[(b^2 + 100 - c^2) / (20b)] : the 100 is 10^2 (side BC).

    It's sure not an easy problem! Good luck.
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  3. #3
    Senior Member pacman's Avatar
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    thanks for the reply, but i have done that already. Like isolating triangle ABD, and summing up their angles = 180 degrees. But the answer was an error . . . . Why not make a complete step by step solution for this? It will frustrate you. Maybe I was wrong with my steps but, that is it. It made some cracks in my skull, emotionally. Where's Grandad?
    But i will rework my busted solution, your suggestion maybe right . . . .thanks though
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  4. #4
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    Didn't get a way to solve directly...yet...maybe there isn't one...

    Let AB = b, AC = c, AD = d, average speed = x.
    Using iteration (7digit accuracy):
    x = 39.1496625, b = 12.8373031, c = 19.5748313, d = 16.0997750

    Program method I used:
    vary x
    b = (7x - 120) / 12, c = x / 2, d = (2x - 30) / 3
    get angle A using triangle ABD [this uses the hypotenuse 10sqrt(2)]
    get angle B using triangle ABC: B = B - 45
    get angle D using triangle ACD: D = D - 45
    stop when (A + B + D) * 10^7 = 180 * 10^7
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  5. #5
    Senior Member pacman's Avatar
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    Wilmer, thanks for giving time to this type of problem. Do you mean that this problem can not be solved directly but by ITERATION? Wow! i will try your method, thanks for the help, i hope i can get something . . . .your answer x = 39.1496625 km/h is reasonable enough.
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  6. #6
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    Quote Originally Posted by pacman View Post
    > Do you mean that this problem can not be solved directly but by ITERATION?

    I said I didn't THINK it can be solved directly!

    > your answer x = 39.1496625 km/h is reasonable enough.

    That is the PRECISE answer to 7digit accuracy (not just reasonable!)
    .
    Last edited by Wilmer; July 27th 2009 at 08:16 PM. Reason: none
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  7. #7
    Senior Member pacman's Avatar
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    Thanks Wilmer.
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