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Math Help - Volume of Wine Glass

  1. #1
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    Lightbulb Volume of Wine Glass

    When a shape lying entirely on one side of the X axis is rotated about that axis it generates a solid of revolution. This investigation explores how solids of revolution and integration can be used to determine the volume of a wine glass.

    Volume of Wine Glass-photo-1.pngVolume of Wine Glass-photo-2.pngVolume of Wine Glass-photo-3.pngVolume of Wine Glass-photo-4.pngVolume of Wine Glass-photo-5.png
    I have attached photos of what I am struggling. It would be amazing if someone could help me out, showing full working and explanations!

    Thanks!
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  2. #2
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    Re: Volume of Wine Glass

    We can help you with parts 1 and 2 if you tell us where you're having difficulty. Assuming you've done parts 1 and 2 with no difficulty, in part 3 you can probably fit a quadratic curve (y = ax^2 + bx + c) to the given shape. You'll need to reserve one parameter for the size - you'll calculate that one after you integrate (so the volume is correct).

    - Hollywood
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  3. #3
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    Re: Volume of Wine Glass

    For part 3, i've found a function for the curve. its -0.003578x^4+0.0962x^3-0.9892x^2+0.333x+0.2634
    How do i find the volume of the wine glass that it creates?
    Thanks
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  4. #4
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    Re: Volume of Wine Glass

    Interesting choice of function. To find the amount of wine it will hold, you integrate \pi{y}^2 - so V=\pi\int_a^b (-0.003578x^4+0.0962x^3-0.9892x^2+0.333x+0.2634)^2 \,dx, where x=a is the bottom of the bowl of the wine glass and x=b is the rim of the wine glass.

    To find the volume of the wine glass, you're finding the surface area and multiplying by 2mm. I'm not sure what you're supposed to do about the stem and the bottom. To get the surface area, you integrate 2\pi{y}\sqrt{1+\left(\frac{dy}{dx}\right)^2} - the first part is just the circumference of a circle with radius y, and the square root is there because a more slanted disk has more surface area. You'll have to do this integral numerically.

    - Hollywood
    Last edited by hollywood; July 11th 2014 at 06:49 AM. Reason: latex
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  5. #5
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    Re: Volume of Wine Glass

    I've changed the function to -0.000962x^4+0.0277x^3-0.337x^2+1.85x+0.0685
    with that, how do i do part 3 question 3?
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