A Tunnel Through the Earth
I have a very short problem this week: How deep will you go if you dig a straight tunnel through the earth, how long will it be, and what angle do you have to start at?
Combinatorics: Multiple Methods, Subtle Wording
(A new question of the week) With few new questions of general interest available this week, I thought I’d go back a few months to a couple little questions on a topic we haven’t dealt with lately, combinatorics. We’ll have one question each on permutations and combinations, showing some subtlety in both the methods we …
A Hole in a Cube
(A new question of the week) Here is an interesting little question about how drilling a hole affects volume and surface area. We’ll have one answer, and several explanations.
Two Tricky Questions on Tangent Lines
(A new question of the week) Sometimes we have lots of quick questions and a number of long discussions, neither of which seems suitable for a post. This time I’ve chosen to combine two distantly related questions, one recent and one from several months ago, both involving tangent lines to functions.
Two Sinusoidal Models
(A new question of the week) Two recent questions involved using trigonometric functions to model real-life (or nearly so) situations, one about breathing, the other about a Ferris wheel. Both can be done by writing a sinusoidal function; the second can be done in other interesting ways as well.
Two Worlds of Relations
Terminology and definitions in mathematics sometimes vary according to context. Here we’ll look at the ideas of relations, functions, and their domains, and discover that they look different from different perspectives.
Trig Identities: Where’d That Come From?
(A new question of the week) Proving a trigonometric identity can be a challenge; sometimes even when we read someone else’s proof, we can fail to see how they came up with a seemingly magical step. We’ll look at two such identities here, and consider how to bridge a gap when we are stuck.
Function Transformations Revisited (II)
(A new question of the week) Last week we examined how a series of transformations affects the equation of a function, in order to write the equation from a graph, or vice versa. We touched on why it works the way it does, but this is something you need to look at from multiple perspectives …
Function Transformations Revisited (I)
(A new question of the week) Transformations of functions, which we covered in January 2019 with a series of posts, is a frequent topic, which can be explained in a number of different ways. A recent discussion brought out some approaches that nicely supplement what we have said before. Here, the focus will be on …
Solving a Triangle: What Went Wrong?
Trigonometry can be a powerful tool for solving sides and angles in triangles. But you have to be careful with it! We’ll look at a classic type of error in solving an SSA triangle, get three explanations, and then see how knowing the context of a question can change our answer – to the point …