Volume of a Cone
Date: 5/9/96 at 19:4:26 From: Anonymous Subject: Volume of a Cone Hello Dr. Math, I was wondering if you could help me with a geometry problem that I have to do: A coffee pot with a circular bottom tapers uniformly to a circular top having radius half that of the base. A mark halfway up (by height) says 2 cups. If the pot could be filled completely to the rim, how much coffee would it hold? Any information on how to solve this problem would be greatly appreciated! Thanks. Joe, 9th grade, Lake Braddock Secondary School.
Date: 12/11/96 at 01:02:47 From: Doctor Rob Subject: Re: Volume of a Cone Good problem! Start by cutting a cross-section through the centers A and C of the circular top and bottom of the pot. Then extend the side lines DF and its counterpart until they meet at P. Connect the two centers A and C with another line, and extend it until it meets the extensions of the side lines at the same point P: P /|\ / | \ / | \ / A| \ /----|----\D Top / | \ / | \ / B|-------\E 2-Cup Line / | \ / | \ ----------------------- Bottom C F Then CF = 2*AD, and AB = BC, so DE = EF, using similar or congruent triangles. Now you can show that PC = 2*AC = 4*BC using similar triangles. You can compute the volume of the cones with vertex at P and bases with centers at A, B, and C, using the formula for the volume of a cone, V = (Pi*h*r^2)/3, where h is the height and r the radius of the base. The volume of the large cone V(L) minus the volume of the medium-sized cone V(M) will give the volume of 2 cups as marked on the pot. The volume of the large cone V(L) minus the volume of the small cone V(S) will be the volume you seek, call it x, measured in cups. The equation is: V(L) - V(S) x ----------- = - V(L) - V(M) 2 which you will then solve for x. All the lengths such as BC and CF from the diagram will cancel from the top and bottom of the fraction on the left in this equation, so they need not be known. I hope this helps. If you still need more help, write back and we'll give it another try. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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