Help with integrals and volume

gravethief13

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Jan 9, 2012
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The problem is to find the height at which a coffee cup is
cramster-equation-2012191810566346172945608124054992.gif
full. The height of the cup is 4.375", the top diameter is 3.5", the bottom diameter is 2", and the side length is 4.6477"

I solved for the volume using the equation
cramster-equation-2012191816226346172978290872567544.gif

where h is height, R is top radius, and r is bottom radius. I got a volume of 26.6299 in3. I multiplied that by
cramster-equation-2012191810566346172945608124054992.gif
to get a volume of 19.9724 in3. From here i'm stuck. I can't solve for h with the new volume because my top diameter is also changeing. Does anyone have any ideas of where to go from here?
 
It is not a question of where to go. It is a question of where to start. You have not specified the actual shape of the cup.

Top Radius = 3.5 / 2 = 1.75
Bottom Radius = 2 / 2 = 1

Consider the right, circular cylinder from the bottom. The cross section of the portion outside the cyllinder has what shape? Let's see if it's a triangle:

\(\displaystyle (1.75-1.00)^{2} + 4.375^{2} = 0.5625 + 19.140625 = 19.703125\)

\(\displaystyle \sqrt{19.703125} = 4.4388 \ne 4.6477\)

I have to believe the outer edge is a curve of some sort. That cross section is not a triangle.

okey-dokey, what is it, then?
 
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