Real life

Finding Length of a Roll: Facing Reality

Last time we looked at how to find the length of material on a roll, making some necessary simplifications. Here, I want to look at some variations on that: first, about carpet in particular, and then about wire on a spool.

Finding the Length of a Rolled-up Carpet

One of the more frequent questions we had on Ask Dr. Math was about how to find the length of material (carpet, paper, wire, etc.) on a roll, knowing only the inner and outer diameters and something else: the thickness of the material, or the number of turns, or the original size of the roll. …

Law of Sines vs Law of Cosines: Which is Better?

Last month, four students from the same class wrote to us with the same question: Which is more accurate, the Law of Sines or the Law of Cosines? Those led to a couple deeper discussions, as we explored the context.

Probability of Consecutive Numbers in a Lottery

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.

Diluting a Solution: Math vs. Reality

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.

The Book Stacking Problem

(An archive question of the week) A recent question asked about a well-known problem about stacking books (or cards, or dominoes) so that the top one extends beyond the base, giving a link to one of many explanations of it – but one, like many, that doesn’t quite fill in all the details. Doctor Rick …

(A new question of the week) Real life questions of probability often require information that we don’t have – they become a job for statistics instead. But sometimes just trying some plausible numbers, as in a Fermi problem, can yield interesting results. Here we consider the probability of an injury when kids play near a …

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?

Distances on Earth 3: Planar Approximation

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.

Distances on Earth 2: The Haversine Formula

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 …