## 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?

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?

We’ve looked at two formulas for the distance between points given their latitude and longitude; here we’ll examine one more formula, which is valid only for small distances. This is a “flat-earth approximation” to distance.

Last week we started a series about finding distances on a sphere (which approximates the shape of the earth), using a straightforward formula from spherical geometry. But in practice, that formula turns out not to be ideal, so a different formula is used when accuracy in all circumstances matters. That is this week’s topic: first …

Many students study trigonometry, but few get to spherical trigonometry, the study of angles and distances on a sphere. This is particularly useful in dealing with measurements on the earth (though it is not a perfect sphere). In this series, we will derive and use three different formulas for the distance between points identified by …

We’ve looked at how to find the circumference of the earth, and how far we can see over the horizon. Another kind of question we’ve had about the curvature of the earth is, how much does it curve over a given distance? That has been asked in several different ways, which lead to some intriguing …

We have been looking at questions about the roundness of the earth, starting with the general fact, and then the determination of the size of the earth. A very common question is about how that roundness affects what we can see, sometimes as a challenge (“If I can see this, then how can the earth …

Last time we looked at a couple questions about proving the earth is round, which led into questions about how Eratosthenes measured the earth (though that in itself did not prove the earth is not flat). Let’s look at two questions about that project itself. Eratosthenes for third graders The first is from 1995, and …

Can you use mathematics to prove that the earth is round? That’s a question we get from time to time, sometimes from people who want to prove the earth is flat, sometimes from people who want to convince their friends otherwise, and sometimes just from students. Let’s think about it. Can we prove it using …

(A new question of the week) Having just discussed several mathematical topics that lie behind the various graphs we have seen in the news lately, I want to depart from our usual style and answer my own current questions. We’ll look at several graphs of COVID-19’s growth and think about what we can learn from …

We’ve been looking at the math underlying some of the graphs associated with the COVID-19 pandemic, starting with exponential growth, and then logistic growth. I want to look in more detail at a feature I mentioned in the first post, viewing a graph logarithmically. This is a powerful technique that goes far beyond a button …