Ant Cone
Here we are going to play with a circular cone of height 16 and base radius 4. First we get each student (or group) to construct a full-size model of the cone (it’s a fun task) and then we introduce an ant that is going to crawl up and down on the cone.
From here we go on to examine the contour diagram of the cone—a set of circles centred at the origin.
Now we turn our attention to the ant who is wandering around on the cone. We keep track of the journey with its path, its projection onto the base of the cone. Our main question is to calculate the ant’s point of highest altitude.
For example, consider the path y = 4 − x. The students will find different ways to argue that the maximum point is (2, 2).
The ant’s journey in 3-space is the curve where the plane x + y = 4 intersects the cone. Is it a parabola?
