Exploration idea #496
Today in Maths HL class we were using a model of a regular tetrahedron (right triangular pyramid with four faces that are equilateral triangles) to help visualize a particular problem in trigonometry. After class was finished, one of the students approached me with an interesting thought. He was wondering at what location would a plane parallel to the base of a pyramid need to intersect the pyramid such that it divided the pyramid into two solids of equal volume. The two solids created by 'slicing' a pyramid in such a way would be another pyramid (similar to the original) and a truncated ('cut off') pyramid - also known as a frustum.
Our discussion only lasted a couple of minutes but we eventually came up with something that I thought could be the start to a genuinely interesting mathematical exploration - which could be the focus of an Exploration (internal assessment task). Here is the gist of it ...
Given a pyramid with height and base area , find the height of the truncated pyramid so that the volume of the truncated pyramid is one-half of the volume of the original pyramid. And further, for this value of what would be the ratio of the base area of the smaller 'top' pyramid to the base area of the original pyramid to the. And lastly, would any pattern result from determining the same ratio for other pyramids - in particular, pyramids with a base that is a regular polygon.
That was where our brief discussion finished today ... but there are several other possible directions in which this investigation / exploration could go. For example: (1) what is the formula for the volume of a truncated pyramid and what parameters is it best written in terms of; (2) what (if any) relationship exists between the ratio of to and the ratio of the bases areas and ; or between the ratio of to and the ratio of the volumes of the 'lower' truncated pyramid to the 'smaller' (or 'upper') pyramid.
And after school was finished for the day and I thought about it some more, it came to me that another direction (and a good one for a Maths HL student Exploration) is to invert the pyramid (upside down) and consider water being added into this solid at a constant rate. What would be the rate of change of the height of the water (height of the 'smaller' pyramid) at the point at which the solid is half full? And how does the rate of change - at this height where the volume is one-half of the entire pyramid - change for pyramids with different bases (e.g. square, regular pentagon, regular hexagon, etc)?
Certainly there are some elements of an Exploration (more than one?) lurking amongst these questions.