We’ve looked in the past at volumes and surface areas of familiar geometric shapes like spheres, pyramids, and cones; but more can be done. If we cut parallel to the base of a pyramid or cone, the result is called a frustum (no, not a frustrum!). Let’s derive some formulas, which will be remarkably simple.
A recent series of questions from an insightful high school student about word problems, provided a number of opportunities to discuss how to find and correct your mistakes – or the book’s! We’ll look at five.
When we recently looked at the Chain Rule, I considered including two questions about its proof, but decided they would be too much. However, when a recent question asked about a different version of the same proof, I decided to post all three. It is a nice illustration of how a mathematician’s view of a …
Last week we looked at the probability that one of infinitely many possible triangles is an acute triangle, and ended up thinking about continuous probability distributions in general. I thought it might be good to look at some old questions dealing with similar issues in various ways. One will even come back to that same …
Some time ago we looked into the probability that a random set of sides (from, say, a broken stick) form a triangle. A recent question asked about the probability that a random triangle is acute (all angles acute) or obtuse (at least one angle obtuse), which led to more discussion of what it means for …
A recent question about lottery numbers reveals that a seemingly special event is in fact surprisingly common: namely, the presence of consecutive numbers in a lottery drawing. The calculation is an interesting one, and we’ll also see a way to check our answer, then compare it to reality.
In recently discussing Roman numerals, we ran across Egyptian multiplication. An improvement on that method is called the Russian peasant method, and deserves attention.
Have you ever wondered how to add, subtract, multiply, and divide using Roman numerals? On one hand, we’ll give the simple answer that the Romans didn’t actually do what you think; on the other hand, we’ll consider what they actually did.
Roman numerals are very different from the “Arabic” system we use; there is no “place value”. And yet, as we’ll see, the two systems have more in common than you might think.
(A new question of the week) Having just discussed the Chain Rule and the Product and Quotient Rules, a recent question about implicit differentiation (which we covered in depth two years ago) fits in nicely. This raises an important issue: when you get an apparently wrong answer, you may just have done something wise that …
Implicit Differentiation: What to Do When It’s “Wrong” Read More »