KJW

The two [math]x[/math]-values of the intersections are: [math]1 - x^2 = y^2 \\ \sigma^2 (5 - x)^2 = 25 \sigma^2 - 10 \sigma^2 x + \sigma^2 x^2 = y^2 \\ (25 \sigma^2 - 1) - 10 \sigma^2 x + (\sigma^2 + 1) x^2 = 0 \\ x = \dfrac{10 \sigma^2 \pm \sqrt{100 \sigma^4 - 4 (\sigma^2 + 1) (25 \sigma^2 - 1)}}{2 (\sigma^2 + 1)} \\ = \dfrac{10 \sigma^2 \pm \sqrt{4 - 96 \sigma^2}}{2 (\sigma^2 + 1)} \\ = \dfrac{5 \sigma^2 \pm \sqrt{1 - 24 \sigma^2}}{\sigma^2 + 1}[/math] where [math]\sigma = \tan\theta[/math], and [math]\theta[/math] is the angle between the surface of the cone and the central axis of the cone.

I often say to people "nobody knows chemistry". This hyperbole comes from my observation that quiz show contestants who seem to know about physics, mathematics, biology, astronomy, geography, history, literature, etc seem to not know about chemistry, passing or getting wrong rather easy questions. It seems to be the one subject that very few contestants know about.

Of course it is possible, but how likely it is I cannot say. However, prior to my first post, I did look at the link to check if this question was indeed part of a maths test rather than simply a textbook exercise, and could find no indication of which it is. I checked because I wanted to know the cost to the student of getting the question wrong, and therefore the importance of having been given the definition prior to the question.

I don't know why it would matter, unless they were taught something wrong. It matters because the student was expected to know the correct answer and therefore should have been taught how to correctly answer the question. As I see it, answering the question is simply a matter of knowing a definition, and therefore marking the student wrong if the definition was never taught is wrong. How many people know that a circle is just the perimeter and doesn't include the area inside? Or that a sphere is just the surface and does not include the volume inside? It seems like a subtle point to require the student to know unless it had been explicitly taught. And if the student was taught this, then the student should have correctly answered the question. But here we are debating the answer to the question (which is simply based on a definition) without the context of what the student was taught.

Deleted.

https://en.wikipedia.org/wiki/Solid_of_revolution

One thing that was not mentioned was what the student was taught prior to being given this question.

Presumably, this means that sodium hydroxide should be added to the tartaric acid first, then potassium hydroxide. I've never seen the dissolution of sodium hydroxide in water be violent, although it does get quite hot. Perhaps adding water to powdered sodium hydroxide is violent, but who uses powdered sodium hydroxide (it's normally in pellet form). Perhaps the KOH (and NaOH) should be assayed by titration.

You could also use hot water. Hot water is better than warm water. 🤪

A difficulty with pool leaks is distinguishing between leaks and evaporation. In my case with a salt-water pool, I replaced the lost water with fresh water until the electrolytic chlorinator stopped full output, indicating a leak and not evaporation (salt doesn't evaporate).

Did you check the waste-water line from the filter (causing loss of water when the pump is on)? This was the first thing the professional checked.

I had a pool leak about a year ago. I called in a professional to locate and fix the leak(s). One of the things he did was to get in the pool under the water and place drops of dye near the places where the leak was most likely to be, carefully observing any movement of the dye that would indicate a leak.

For arbitrary function [math]\psi[/math], and [math]x^\alpha[/math] and [math]\dfrac{\partial}{\partial x^\beta}[/math] as operators: [math]\dfrac{\partial}{\partial x^\beta} \Big(x^\alpha \psi\Big) - x^\alpha \dfrac{\partial}{\partial x^\beta} \psi = \delta^\alpha_\beta\ \psi[/math] Thus: [math]x^\alpha[/math] and [math]\dfrac{\partial}{\partial x^\beta}[/math] are conjugate variables (operators) for [math]\alpha = \beta[/math]. Note that: [math]p_\gamma = -i \hbar \dfrac{\partial}{\partial x^\gamma}[/math] So that: [math][x^\alpha,p_\beta] = x^\alpha p_\beta - p_\beta x^\alpha = i \hbar\ \delta^\alpha_\beta[/math]

No, they are not.

You need to post something here that can be discussed without visiting an external website.

That's "un-American"! And I say "un-American" because I can't think of any other country that thinks like that. Yes, capitalism exists throughout the West, and conservative governments do push for privatisation, but this idea that any hint of socialism is thoroughly evil seems to me to be a uniquely American phenomenon.

I'll concede that point. On the other hand, you did say that "a clue should be the use of 'classical' in the name", indicating that it is obvious that the classical electron radius is not the actual size of the electron, and that it was not necessary to explicitly state this. However, I do think that the classical electron radius is in some sense an effective radius or size scale of a dressed electron. Where would one expect the size of an electron to appear in physics, anyway? It isn't as though the Bohr radius is used in physics that isn't about the size of atoms. The classical electron radius appears in non-relativistic Thomson scattering and the relativistic Klein–Nishina formula.

This may help: https://en.wikipedia.org/wiki/Periodic_table

I'm not an American, so I can't answer this question. However, I did have a rather strong interest in this US election, more than any previous US election (and even more than the recent state election at home). Based on the information I was receiving, it seemed to me that Trump was going to win. Although the political commentators were saying the polls were within the margin of error, the polls were nevertheless pointing to a Trump victory. And many of the "ordinary folk" that I saw interviewed were saying that they were voting for Trump. But I'm no expert and certainly wouldn't have locked in a prediction. Watching the election results come in, it did seem like a foregone conclusion from the start.

The flaw with this is that elections are not random and therefore applying probability is not valid. Some elections are easy to predict the outcome, others not so easy. If some methodology correctly predicts easy elections but gets the hard ones wrong, then that methodology is really quite useless, regardless of how many easy elections are correctly predicted.

The way I see it, different people have different ideas about how their country should be run, and therefore it is not possible to please everyone, regardless of the political system. That's not to say that all political systems are equal in terms of benefiting the population. But to think that one can solve the problems inherent in political systems simply by providing more democracy is misguided due to the fundamental differences within the population. There is such a thing as "tyranny of the majority".

What you said (to which I replied) was that size doesn't apply to fundamental things like an electron, whereas the classical electron radius is a size that does apply to the electron. That it was derived using classical electrodynamics is beside the point. It is not being suggested that the classical electron radius is the actual radius of the electron, but in some sense, it is an "effective" radius of the electron and a physical constant that appears in theories where the size of the electron is relevant. Another size associated with the electron is the Bohr radius. Even though this was derived using an obsolete model, it remains as a physical constant that appears in modern theories regarding the size of an atom. See above. I didn't actually say that the classical electron radius is the actual size of the electron. I referred to a Wikipedia article that describes the classical electron radius as "a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation".

This is reminiscent of the crystallisation of a racemic mixture, which can either crystalise as the racemate or as a mixture of enantiomeric crystals. This is especially relevant to this thread because the first resolution of a racemic mixture was of sodium ammonium tartrate by Louis Pasteur who manually separated the individual enantiomeric crystals into separate piles. It is said that he was quite fortunate to have found a racemic mixture that crystallises as a mixture of enantiomeric crystals because most racemic mixtures crystallise as the racemate.

https://en.wikipedia.org/wiki/Classical_electron_radius

Seems like a statement that there are conserved quantities in physics. Hardly groundbreaking.