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.
OK, here are my 2 cents
Only the second part of the challenge, the proof.
Prime 10 was able to handle this almost entirely on its own, only a little help from legacy Mathcad was needed in one place.
Maybe an incentive to improve Primes Symbolics a little further in this direction... 😉
Nice work!
It's also easy enough to get a symbolic solution to:
by hand, without resorting to another CAS program:
Alan
It's also easy enough to get a symbolic solution to:
Agreed on - especially because both equations simplify to equations in just one variable each.
It was surprising that Prime could not deal with that situation.
by hand, without resorting to another CAS program:
"by hand", that's the point - my goal was not just using Prime as an equations editor (MathType would do a much better job if that was all it was about) but to let the program do the (not so hard) work on its own (and of course I couldn't resist to show that Mathcad can do it 🙂 ).
BTW, even Prime is able to solve each of the equations on its own if we help it by telling it which variable to solve for.
Forcing the solution in the needed range from 0 to pi/2 seems to confuse Prime - the solution of the second equation is correct but written in a very unusual way:
adding "simplify" or even "rewrite, asin" did not help.
Alright, here is my entry. Please enjoy (if possible;).
I am a structural engineer so I might tend to skip steps than in a more rigorous, academic proof. Generally, those that review my engineering work are very familiar with the concepts I present, so I usually go for the simplest consistent formatting—I do not want to miss anything. And I add plots whenever I can so that I can tell if something is not quite right.
I hid much of the grunt work for the plots, you can reveal them easily enough in the Mathcad sheet.
Here are sample screen shots:
