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Thread: General Formula For Variation 1

  1. #1
    Senior Member harpazo's Avatar
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    General Formula For Variation 1

    Write a general formula to describe each variation. Let k be the constant of proportionality for each variation.

    1. v varies directly with t.

    v = tk

    2. V varies directly with x^3.

    V = (x^3)k

    3. y varies inversely with sqrt{x}.

    y = k/sqrt{x}

    Is any of this correct?
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    MHF Contributor MarkFL's Avatar
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    Re: General Formula For Variation 1

    Yes, they are correct, although in the first 2, I would write k as the first factor, as this is the general convention, to put the parameter before the variable.
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    Senior Member harpazo's Avatar
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    Re: General Formula For Variation 1

    Quote Originally Posted by MarkFL View Post
    Yes, they are correct, although in the first 2, I would write k as the first factor, as this is the general convention, to put the parameter before the variable.
    What exactly is meant by constant of proportionality?
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    MHF Contributor MarkFL's Avatar
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    Re: General Formula For Variation 1

    Quote Originally Posted by harpazo View Post
    What exactly is meant by constant of proportionality?
    A constant of proportionality, which is denoted many times by the letter k, represents some positive real number. For example, we could say that the area of a circle varies as the square of its radius:

    $\displaystyle A=kr^2$

    We recognize that in this case we have:

    $\displaystyle k=\pi$

    The area of a circle will also vary as the square of its circumference and diameter too. Any linear aspect of the circle we choose, the area will vary as the square of this aspect.

    The area of a two dimensional place figure will always vary as the square of one of its linear measures. For example, if we have two similar figures and the larger figure is found to have a certain linear measure that we find to be twice as long as the corresponding measure in te smaller figure, then we know the area of the larger figure is 4 times that of the smaller. This means, as an example, that if we have a circle of radius r, then a circle having a radius of 2r will have an area 4 times that of the smaller circle.
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    Senior Member harpazo's Avatar
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    Re: General Formula For Variation 1

    Quote Originally Posted by MarkFL View Post
    A constant of proportionality, which is denoted many times by the letter k, represents some positive real number. For example, we could say that the area of a circle varies as the square of its radius:

    $\displaystyle A=kr^2$

    We recognize that in this case we have:

    $\displaystyle k=\pi$

    The area of a circle will also vary as the square of its circumference and diameter too. Any linear aspect of the circle we choose, the area will vary as the square of this aspect.

    The area of a two dimensional place figure will always vary as the square of one of its linear measures. For example, if we have two similar figures and the larger figure is found to have a certain linear measure that we find to be twice as long as the corresponding measure in te smaller figure, then we know the area of the larger figure is 4 times that of the smaller. This means, as an example, that if we have a circle of radius r, then a circle having a radius of 2r will have an area 4 times that of the smaller circle.
    Thank you very much. Also, thank you for your patience as I go through the textbook alone.
    Thanks from MarkFL
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