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Math Help - Algebra (Volume, Ratios)

  1. #1
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    Algebra (Volume, Ratios)

    The figure (fig) shows a solid consisting of three parts, a cone, a cylinder and a hemisphere, all of the same base radius.

    The first part of the question asks us to find the volume of each part, in terms of w, s, t and π (pi):

    Volume of cone = (1/3)πt2w

    Volume of cylinder = πt2s

    Volume of hemisphere = (1/2)*(4/3)πt3

    The next part of the question reads:

    i: If the volume of each of the three parts is the same, find the ratio w : s : t.
    ii: If also w + s + t = 11, find the total volume in terms of π.

    Aaaaaaaand I have no idea, nor am I expected to have an idea -- I was never actually taught how to do this, it was some kind of maniacal challenge set by my teacher to see how I would go about finding an answer.

    Any help would be much appreciated. Thanks for your time.
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  2. #2
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    Re: Algebra (Volume, Ratios)

    Well start by setting two of the volumes equal to each other.

    \displaystyle \begin{align*} \frac{1}{3}\,\pi\, t^2 \, w &= \pi \, t^2 \, s \end{align*}

    and find the relationship between w and s. You should be able to form the ratio between them from there.
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    Re: Algebra (Volume, Ratios)

    So I did (1/3)πt2w = πt2s and came out with s = (1/3)w and thus w = s/(1/3). I tried solving (1/3)πt2w = (1/2)*(4/3)πt3 for t but failed miserably.

    Have I done this right? Also, what significance does this have in finding the ratio?
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    Re: Algebra (Volume, Ratios)

    Bump. Anyone?
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    Re: Algebra (Volume, Ratios)

    Algebra (Volume, Ratios)-ratio-volume.png
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    Re: Algebra (Volume, Ratios)

    Quote Originally Posted by ibdutt View Post
    Click image for larger version. 

Name:	Ratio Volume.png 
Views:	2 
Size:	8.5 KB 
ID:	27345
    So how did you go from having (1/3)w = s = (2/3)t to having the ratio (6:2:3)?

    And also, where did "k" come from?
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    Re: Algebra (Volume, Ratios)

    Algebra (Volume, Ratios)-new-4-mar-13.png
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    Re: Algebra (Volume, Ratios)

    Thanks heaps. One final question to clarify my understanding: do we divide by 2 to make each fraction (1/X), (1/Y), etc? (i.e to turn (2/3) into (1/3) we must divide it by 2 and thus all the others by 2)
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    Re: Algebra (Volume, Ratios)

    Yes we divide the fractions by a suitable number so that we have 1 in the numerator of each fraction, in this case we divide by 2.
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    Re: Algebra (Volume, Ratios)

    Thanks heaps.
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