Mathcad Community Challenge November 2024 - Frustums and Pyramids
This month’s challenge is based around pyramids and frustums (a pyramid that’s had its top removed by a plane parallel to the base). Choose any or all of the following and create a Mathcad worksheet:
Moderate Challenge
Calculate the volume and surface area of the frustum in the image below. A Creo 7.0 part model is attached to this challenge for verification.
Note that the original pyramid is a right pentahedron. (A line joining the center of the base and the vertex of the pyramid is at a right angle / perpendicular to the base. See the CAD model and image for additional clarification.) The base and top surface of the frustum are both square.
Optional 1: Create a function that calculates the volume given the 3 dimensions above: edge length of the base, side edge length, and top face edge length.
Optional 2 (for Mathcad Prime 10 users): Use the slider advanced input control to change either the height of the slicing plane or length of the side edge, and recalculate the volume.
Hard Challenge: The Pyramid of Least Volume
“Of all the planes tangent to the ellipsoid
one of them cuts the pyramid of least possible volume from the first octant x ≥ 0, y ≥ 0, z ≥ 0. Show that the point of tangency of that plane is the centroid of the face ABC.”
The pyramid in this situation has four sides. One of the corners is at the origin (0,0,0). The centroid is this case means the intersection of its medians.
(Source: “The Mathematical Mechanic” by Mark Levi, section 3.5.)
Documentation Challenge
Create a Mathcad worksheet that uses text and image tools to explain the derivation of the formula for the volume of a pyramid.