The Mathcad Community Challenge for November 2024 was based around pyramids and frustums (a pyramid that’s had its top removed by a plane parallel to the base). Click to the challenge thread on PTC Community to download all of the Mathcad Prime worksheet submissions.
Calculate the volume and surface area of the frustum in the image below. A Creo 7.0 part model was attached to the challenge for verification.
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.
“Of all the planes tangent to the ellipsoid
x2/a2 +y2/b2 +z2/c2 =1
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.)
Five people submitted a total of six worksheets. The quality in all of them is quite amazing.
Frequent contributor Alan Stevens was the first person to submit a worksheet, and as usual, his worksheet can double as a teaching tool. The worksheet is laid out beautifully with text regions, images, and math regions with the comparison equals to operator to explain the logic. The final values for the volume and surface area are highlighted nicely. Since Alan performed the challenge using Mathcad Prime Express, he used an XY Plot to show volume as a function of the side edge length of the frustum.
User PPal submitted two worksheets, one for each challenge. His also makes excellent use of the documentation tools in Mathcad. The first worksheet uses two different methods to calculate the frustum volume. The first method involves integration and takes advantage of the slider advanced input control in Prime 10. The second method uses a triple integral in a pyramid coordinate system to calculate the volume. Neat stuff.
PPal’s second worksheet tackles the Pyramid of Least Volume. His solution method normalizes the ellipsoid to a sphere and incorporates Lagrange multipliers, derivatives, and symbolic evaluation to prove that the point of tangency is at the centroid of the triangle.
The next worksheet was from Werner, also a frequent contributor. He tackled the hard challenge with a very attractive worksheet. (Turning off the grid, formatting the text, and applying background colors to regions make a huge difference.) Werner walks through his proof using symbolic evaluation and partial derivatives to find the coordinate points of the plane that result in the minimum volume, calculates the center of gravity of its triangle, and sees if that is a point on the ellipsoid. Then he graphs the ellipsoid, triangle, and tangent plane with a little help from the CreateMesh function. Some really advanced work here.
Repeat contributor TTokoro submitted a worksheet that really pops off the page with its use of font color and highlight color. He calculates the angle and edge length for the original pyramid, which allows him to calculate the volume of the frustum by subtracting the volume of the smaller removed pyramid from the original pyramid. He makes use of a short program to facilitate calculation of the surface area. For the optional part, he used both slider and checkboxes to update a 3D Plot of the frustum. The worksheet concludes with a couple XY Plots of volume and surface area as a function of frustum height, including vertical and horizontal markers. It’s great to see a different approach compared to the other submissions.
The final worksheet was from Spauliszyn, a first time contributor. I often comment on aesthetics, because that’s the first thing I typically notice. The lengthy worksheet had nice headers and footers, slider controls, 3D Plots, and collapsed areas that held many of the math regions. This is a fun worksheet to play with, because you can control manual or slider input, with sliders for the lengths of the top, bottom, and side edges, which update the 3D Plot of the frustum.
The solution to the Pyramid of Least Volume also contains a slider control, which I did not expect. It starts with a couple 3D Plots depicting the ellipsoid and tangent plane. The formatting of the proof is simply beautiful, making use of derivatives and symbolic evaluation to prove that the centroid of the triangle coincides with the tangent point.
Once again, we see a variety of approaches that users take to solve the same problem, using a variety of tools within Mathcad. On the math side, we see functions, programs, symbolic evaluation, and matrices. Operators include derivatives, partial derivatives, and even the gradient operator. Advanced input controls include lists, check boxes, and sliders. There were quite a few XY and 3D Plots.
All of this was complemented by extensive use of documentation tools and images, making it so that someone unfamiliar with the problem being solved can understand the approaches and results. I highly recommend that you check out the submissions to see what you can use in your Mathcad worksheets.
Join us in January 2025 for an electrical engineering themed challenge! And learn from the previous history of Mathcad Community Challenges here. You can also subscribe to when new challenges are up from there.
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