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Math Help - Weird rate problem.

  1. #1
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    Weird rate problem.

    A hemispherical bowl has a radius of 3.7m. Water is added to the bowl at the rate of pi/3  m^3/min. When the depth of water in the bowl is h m, and volume of water is given by V= (1/6)pi*h^2(9-2h)m^3. How deep is the water after 2.5 minutes? At what rate is the depth increasing at this time?

    So I'm guessing that the m^3 is just a unit and not part of the formula.

    The volume of water in the bowl is pi/3 * 2.5 = 5pi/6<br />
    So 5pi/6=pi/6* h^2(9-2h)

    simplify: 5=9h^2 - 2h^3

    If I'm right so far how do I solve for h?
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  2. #2
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    Quote Originally Posted by coolguy00777 View Post
    A hemispherical bowl has a radius of 3.7m. Water is added to the bowl at the rate of pi/3  m^3/min. When the depth of water in the bowl is h m, and volume of water is given by V= (1/6)pi*h^2(9-2h)m^3. How deep is the water after 2.5 minutes? At what rate is the depth increasing at this time?

    So I'm guessing that the m^3 is just a unit and not part of the formula.

    The volume of water in the bowl is pi/3 * 2.5 = 5pi/6
    The volume of the water at t= 2.5
    [/tex]
    So 5pi/6=pi/6* h^2(9-2h)

    simplify: 5=9h^2 - 2h^3

    If I'm right so far how do I solve for h?
    That's a cubic equation: 2h^3- 9h^2+ 5= 0.
    There is a general formula but it is very complex. The "rational root theorem" says that any rational roots must be of the form a/b where a divides the constant term, 5, and b divides the leading coefficient, 2. That means that, for this equation, the only possible rational roots are \pm 1, \pm 5, \pm \frac{1}{2}, and \pm \frac{5}{2}. There is no guarentee that this equation has a rational root but I recommend trying those to see. If none of those work, there is no simpler solution than the cubic formula.
    Last edited by HallsofIvy; March 19th 2009 at 06:48 AM.
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  3. #3
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    Wow, did I make a mistake in my conclusion. We haven't used cubic roots in my calculus 1 class.
    Last edited by coolguy00777; March 19th 2009 at 07:28 AM.
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