Ice cream cone math probelm?

Question

You are to place a sphere of ice cream into a cone with the height of 1.
a) What radius of the sphere will give you the most volume of ice cream inside the cone (as opposed to above the cone) for a cone with a base angle of 30 degrees?

b) What percent of this sphere of most volume lies inside the cone?

Things to consider:

This is essentially a complicated max/min problem, and a rather silly application of techniques that are very important in solving more realistic problems in which one needs to optimize some geometric relationship between unlike objects. You will find this project difficult because you will have to determine from sketches expressions involving more that one variable, and then compute with those expressions. In order to make progress, you will have to choose to let some of the variables act as parameters, at least temporarily.

If you would like a sketch of the drawing please email me... [e-mail]

cone is 1 unit high

Answer 1

First answer: 2*sqrt(3) - 3 = 0.464 if the entire sphere lies within the cone


Well as the sphere does not lies entirely within the cone (because it has got to be largest), so the radius is 2*sqrt(3) = 3.464 [I took height of cone = 1]

Second answer: 1.286% of the sphere lies inside the cone

Answer 2

What has a height of 1? Is the ice cream only 1 unit above the top of the cone? Or is the cone 1 unit high? Or is the ice cream i unit below the top of the cone?
It seems the upper limit is the ice cream diameter the same as the diameter of the cone, at 2 units. The lower limit will be a sphere where the diameter is at the depth of it's radius. Find the change of volume with respect to the change of diameter and the 1st derivitive should get you your answer.

Answer 3

Ice cream on cones are greater useful,extremely. The sandwhich part of the ice cream sandwich is in basic terms too yummy,you do no longer word the ice cream. additionally,it extremely is exciting slurping ice cream from the tip of the cone. Yayz!