Mathematicians have been slicing open cones since the 4th Century BC. Here, we study one of these famous cross sections — the ellipse. Along the way, we’ll discover a beautiful proof from Germinal Dandelin of the 19th Century.
Slice a right circular cone at an angle to the horizontal. The cross-section forms an ellipse.
Figure 1 shows a right triangle parachuting to the ground with a bedsheet.
Here is a neat application of Bayes’ Theorem. Imagine that a professor gives her students an exam. She wants to know how likely it is than any of them cheated. But she can’t just ask them. Can she even expect students to be report honestly on an anonymous survey? (Is it really anonymous? Really?)
I recently posted this on Reddit:
I am working on an article on Bayes Theorem and statistical significance, and I am asking for your help, dear Redditors! Here’s what I need you to do:1. Flip a coin.2. If the coin turns up…
Last week, Dr. Peyam (of YouTube fame) published a video of a geometry problem he found on Instagram. The solution, he says, “uses a lot of beautiful geometry and even more beautiful calculus.” (It does indeed. Check it out.)
I am going to attack the same problem using methods that predate calculus — and even algebra. Here goes!
A parabola formed by the graph of y=x² cradles a circle of radius 1. The circle touches the parabola at exactly two points. What are the coordinates of the centre of the circle?
You have two circular washers. Each is made of the same material; each has the same thickness. The central holes of the two washers have different diameters.
On each washer, a straight line is drawn from one edge to the other, so it just touches the central hole. The two straight lines each have the same length (Figure 1).
The Secretary Problem is classic Decision Theory scenario. You must choose one of N possible candidates. You have an objective way of ranking them. The proviso is that after you examine a candidate, you must either choose or reject that candidate. There is go going back for a second look.
For example, in Princess Alice and the 1,001 Suitors, once Alice met a potential husband, she either decapitated him or married him.
What strategy will maximize the probability of choosing the best candidate?
Suppose you are at an automobile auction. Each of 19 cars will be displayed, one at a…
Here’s the challenge. You have three infinitely-sided dice. When you roll one of these dice, you get a Real Number between 0 and 1. So when you roll three of these dice, you get a Real Number between 0 and 3. Capish?
Here’s the question. You roll the three dice. You square the outcome of each die.
What are the odds that the sum of the three squares will be less than or equal to 1?
These examples should clarify: