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Math Help - How to simplify velocity magnitude equations?

  1. #1
    Junior Member
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    How to simplify velocity magnitude equations?

    In my calculus class a few days ago, my professor was going over how to find three dimensional velocity and its magnitude. In the problem, the magnitude of the velocity came out to be \sqrt{9+36t^2+36t^4}=3+6t^2.

    My question is, how did he simplify the equation under the square root so that it became 3+6t^2?

    For the homework, I need to simplify these two square root equations to get something without a square root sign (if possible). Can you guys help me out with this? Thanks.
    1) \sqrt{9t^4+36+36t^2}

    2) \sqrt{(-3cost)^2+(-4cost)^2+(-5sint)^2)}
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  2. #2
    Senior Member
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    Hello Infernorage!

    Quote Originally Posted by Infernorage View Post
    In my calculus class a few days ago, my professor was going over how to find three dimensional velocity and its magnitude. In the problem, the magnitude of the velocity came out to be \sqrt{9+36t^2+36t^4}=3+6t^2.
    Do you know this rule:

    (a+b)^2 = a^2+2ab+b^2

    It is not difficult to see that 9+36t^2+36t^4 = (6t^2+3)^2

    So

    \sqrt{9+36t^2+36t^4}=3+6t^2 = \sqrt{(6t^2+3)^2} = 3+6t^2


    Quote Originally Posted by Infernorage View Post
    My question is, how did he simplify the equation under the square root so that it became 3+6t^2?

    For the homework, I need to simplify these two square root equations to get something without a square root sign (if possible). Can you guys help me out with this? Thanks.
    1) \sqrt{9t^4+36+36t^2}
    This one is tricky.
    I told you (a+b)^2 = a^2+2ab+b^2. You can't use it here unless :

    \sqrt{9t^4+36+36t^2} = \sqrt{9*(t^4+4+4t^2)}

    Hence

     \sqrt{9*(t^4+4+4t^2)} = \sqrt{9(t^2+2)^2}


    Quote Originally Posted by Infernorage View Post
    2) \sqrt{(-3cost)^2+(-4cost)^2+(-5sint)^2)}
    It is sin^2+cos^2=1

    and furthermore

    \sqrt{(-3cost)^2+(-4cost)^2+(-5sint)^2)}

    =\sqrt{9cos^2t+16cos^2t+25sin^2t}

    =\sqrt{25cos^2t+25sin^2t}

    =\sqrt{25(cos^2t+sin^2t)}

    =\sqrt{25*(1)} = \sqrt{25}

    Do you understand?

    Yours
    Rapha
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