Volume of cylinders and cones

[theatergirl01]

A cone has diameter 12 and height 9. A cylinder is placed inside the cone so the base of the cylinder is concentric with the base of the cone and the upper base of the cylinder is contained in the surface of the cone. If the volume of the cone is nine times the volume of the cylinder, find the dimensions of the cylinder.

So far, I have determined that the volume of the cone is 108pi, and once divide by 9, the volume of the cylinder is 12pi. Now, I am stuck. Help please!

 

[tkhunny]

1) I am suspicious of this problem. We're not maximizing anything. We just picked a volume. I have a hankering to keep my eyes open for evidence that an answer might not be unique.

2) You are almost there. The only thing you have left is getting the corner of the cyllinder on the cone.

Put a half cross section of your cone on a set of Cartesian coordinates.
Put the tip of the cone at (0,9).
Put the edge of the radius of the base of the cone at (6,0)
Connect the two points (0,9) and (6,0)
Write the equation of this line between (0,9) and (6,0). Call this line C.
Pick a point on the x-axis. Pick a point x, where 0 < x < 6. Call this point R.
Draw a vertical line from this point you hae selected. Quit when you line C. Label this point of intersection D.
From point D, draw a horizontal line to the y-axis. Call this point of intersection H.
If Origin, R, D, H defines the half cross sectoin fo the cyllinder, what is the cyllinder's volume?