## Why Do Logarithms Work That Way?

Last time, we introduced logarithms by way of their history. Here, we’ll look at their properties.

Last time, we introduced logarithms by way of their history. Here, we’ll look at their properties.

Last time we looked at how to find the volume of a frustum of a pyramid or cone. But sometimes what looks at first like a rectangular frustum actually isn’t. This case turns out to have a more general formula almost as nice as what we have for an actual frustum. We’ll discover that the …

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 new question of the week) A recent question asked for the connection between two different ways to use determinants geometrically: to find the area of a triangle, and to find the volume of a pyramid (or the area of a parallelogram and the volume of a parallelepiped). Last time we looked at what a …

How Can 3×3 Determinants Give Both Area and Volume? Read More »

Here is a little question about making a formula to dilute a solution; we’ll see how to do the algebra, and also how what we teach in math classes isn’t quite real.

(A new question of the week) Several interesting geometry problems about triangles and circles came in recently. We’ll look at two today, and a third next week.

(An archive question of the week) In preparing the last couple posts, on recurrence relations, I ran across an answer to a much harder question, that illustrates what it can take to solve one that doesn’t fit the convenient forms. It’s linear, but the coefficients are not constant as they have been in all our …

A Challenging Homogeneous Second-Order Recurrence Read More »

Last week we looked at a recent question about recurrence relations, and I realized it needs a companion article to introduce these ideas. So here we will look at some answers from Ask Dr. Math about the simpler case, including general methods, why they work, and applications.

Here is a short discussion of a common type of problem in trigonometry classes: finding a trig function of the sum or difference of two angles, given minimal information about them.

Last time we looked at what a degenerate conic section is, and how it relates on one hand to actual cones, and on the other to the general equation of the conic. Here we’ll look at the parameters of conic sections (focus, directrix, axes, and especially eccentricity) and how they apply to degenerate cases. Does …

Degenerate Conics II: Are Their Parameters Meaningful? Read More »