document.write( "Question 621373: what is the equation required to create a simple wine glass in a graph and how can we find its volume? \n" ); document.write( "

Algebra.Com's Answer #390760 by KMST(5328) $\"\"$ $\"About$

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Since you placed that question under \"Quadratic Equation\", I assume the expected profile is a parabola, the graph of a quadratic function. I inspected all the wine glasses in my china cabinet and in my wine cupboard and concluded that a paraboloid is not the shape of a wine glass. The mouth of the glass is never the widest part. Wine glasses can be shaped like truncated ellipsoids (white wine glasses) or have a truncated egg shape (with the tip cut off). The bulb of wine glasses is not shaped like a paraboloid, but I guess a paraboloid is the simple shape envisioned. (I would suggest a truncated ellipsoid, but you may not have studied ellipses yet).
\n" ); document.write( "I also assume that you are in a Calculus class and were taught how to use integration to find the volume of a solid of revolution.
\n" ); document.write( "If the height above the bottom of the bulb is $\"h\"$ and the cross-section radius is $\"x\"$ , the shape of a paraboloid wine glass bulb would given by
\n" ); document.write( " $\"h=ax%5E2\"$ <--> $\"x%5E2=h%2Fa\"$
\n" ); document.write( "A cross section of the glass is a circle with area $\"pi%2Ax%5E2\"$ ,
\n" ); document.write( "and since $\"x%5E2=h%2Fa\"$ , that cross section area would be $\"pi%2Ah%2Fa\"$
\n" ); document.write( "If the height of the bulb (vertical distance from the rim to the bottom of the bulb) is H, the volume could be calculated as
\n" ); document.write( " $\"V=int%28%28pi%2Fa%29%2Ah%2Cdh%2C0%2CH%29\"$
\n" ); document.write( "If you use centimeters, then you would get the volume in cubic centimeters, which are essentially the same as milliliters.
\n" ); document.write( "I would chose something close to 6-8 cm for the diameter at the top, and 10 cm for the height (H) of the wine glass bulb.
\n" ); document.write( "With a paraboloid shape, and an 8-cm top diameter, you have x=4 and h=10 at the rim, so
\n" ); document.write( " $\"10=a%2A4%5E2\"$ --> $\"a=10%2F16=5%2F8\"$ .
\n" ); document.write( "In that case
\n" ); document.write( " $\"V=int%28%285pi%2F8%29%2Ah%2Cdh%2C0%2C10%29=%285pi%2F8%29%2A10%5E2\"$ = about 196 mL \n" ); document.write( "

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