Pyramid Volume Challenge
Main Concepts
Multivariable Calculus
Partial Derivatives
Tangent Planes
Duration
25 minutes in class
Summary of Activity
Students are given the challenge to find the volume of a pyramid defined as tangent to a certain surface and bounded by the three standard planes, with respect to the coordinates of the point of tangency
Notes and Insights
This problem allows for at least two different solutions depending on how the function F is set up.
A possible extension is to ask the students to set up a similar problem in the two dimensional plane.
Another extension is to consider the problem in reverse. We fix the volume of a tetrahedron and think about the space that the tetrahedron can cover in the first octant if it moves around. That is the notion of envelope, and it requires a lot more tools than what we have in this course.