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Math Help - Adding complicated quotients

  1. #1
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    Adding complicated quotients

    Again, the book has confused me. What on Earth have they done?
    I got a bit further on, and I've got the same answer they reached doing this, except I don't know where their 1/2 went. Theirs doesn't have it and mine does.
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  2. #2
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    Hi !

    Ok, we have :

    S=\frac{\pi r^3}{3 (100-r^2)^{\frac 12}}+\frac{2 \pi r (100-r^2)^{\frac 12}}{3}

    Let \boxed{k=(100-r^2)^{\frac 12}}, so we have :

    S=\frac{\pi r^3}{3k}+\frac{2 \pi r \cdot k}{3}

    What is the common denominator ?
    In one side, we have 3k and in the other side, 3.

    So the common denominator is \boxed{3k}

    To make such a thing, we will have to multiply the fraction on the right side by \tfrac kk so that we will have 3k at the bottom.


    S=\frac{\pi r^3}{3k}+\frac{2 \pi r \cdot k^{\color{red}2}}{3k}=\boxed{\frac{\pi r^3+2 \pi r k^2}{3k}}


    But what is k^2 ?
    A rule of exponents tells us : (a^b)^c=a^{bc}

    Therefore k^2=\left((100-r^2)^{\frac 12}\right)^2=(100-r^2)^{\frac 12 \cdot 2}=(100-r^2)^1=100-r^2


    ---> S=\frac{\pi r^3+2 \pi r (100-r^2)}{3(100-r^2)^{\frac 12}}


    Is it clear enough ? :x
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  3. #3
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    :O It makes sense! That's phenomenal.
    It's like if you were to do:
    2/10 + 1/5
    You would end up with:
    2/10 + 2/10, to get a common denominator...And then you get 4/10. It's just a coincidence that the 1/2 power cancels out and all, and it caused it to slip my mind :P
    BUT, there is one thing left unexplained. The pi symbol that becomes negative. Do you know why that is?

    Forgive me, I don't quite know how to use the math symbols.
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  4. #4
    Moo
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    :O It makes sense! That's phenomenal.
    It's like if you were to do:
    2/10 + 1/5
    You would end up with:
    2/10 + 2/10, to get a common denominator...And then you get 4/10. It's just a coincidence that the 1/2 power cancels out and all, and it caused it to slip my mind :P


    BUT, there is one thing left unexplained. The pi symbol that becomes negative. Do you know why that is?
    Nope
    Mistake ?

    Forgive me, I don't quite know how to use the math symbols.
    Well, it's still understandable ^^

    But you can try learning how to use the latex in this section : http://www.mathhelpforum.com/math-help/latex-help/
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  5. #5
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    Oh, I think a few lines before it was a negative, and then when it wasn't it was a typo.
    So it's negative. :P
    Thank you very much. You've been very helpful.
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