Math CentralQuandaries & Queries


Question from Mary, a student:

I have to use a 8 1/2 inch by 11 inch piece of paper to make a cone that will hold the maximum amount of ice cream possible by only filling it to the top of the cone. I am then supposed to write a function for the volume of my cone and use my graphing calculator to determine the radius and height of the circle. I am so confused, and other than being able to cut the paper into the circle, I do not know where to start. Thank you for your help! -Mary


Once you have the circle, you are intended to make two radial cuts, cutting it into two sectors. One sector will have an angle less than or equal to 180 degrees, the other will have a reflex angle (bigger than 180 degrees.) Either can be rolled into a cone.

If you roll one of these into a cone, the volume is a little tricky to calculate. The key is to note that the circumference of the base of the cone is the arc of the sector; divide that by 2 pi to get the base radius r.

You also need the cone's height. That, the base radius, and the original radius form a right triangle; use Pythagoras' theorem to find h.

Good Hunting!

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