Country Boy

It sounds like you are looking for "x" such that [math]\int_a^x f(t)dt= \int_x^b f(t)dt[/math]. For example, if [math]f(x)= x^2[/math], a= 0, and b= 1, then [math]\int_0^x t^2 dt= \frac{1}{3}x^3[/math] while [math]\int_x^1 t^2 dt= \frac{1}{3}- \frac{1}{3}x^3[/math]. They will be equal when [math]\frac{1}{3}x^3= \frac{1}{3}- \frac{1}{3}x^2[/math]. Multiplying both sides by [math]\frac{1}{3}[/math] and then adding [math]x^3[/math] to both sides, [math]2x^3= 1[/math], [math]x^3= \frac{1}{2}[/math] so [math]x= \frac{1}{\sqrt[3]{x}}[/math].

Let r be the radius of the circle. Then the segment from the center of the circle to AB, with length p, the segment from the center of AB to point B, with length AB/2, and the segment from the center of the circle to B, with length r, form a right triangle with hypotenuse of length r, legs of length p and AB/2. By the Pythagorean theorem, (AB)^2/4+ p^2= r^2. Now do the same thing with chord AC getting (AC)^2/4+ q^2= r^2. Since "AB=2AC", (AB)^2/4= (AC)^2 so we have the two equations (AC)^2+ p^2= r^2 and (AC)^2/4+ q^2= r^2.

Well, I do mean offense to the OP! The original post was nothing but an attempt to get people to click on the links to advertisements. I am surpised (and a little disappointed) that this thread has gone on so long.

Tesla worked for Edison with a contract that assigned all patents to Edison. I suspect whether or not one would be willing to sign such a contract depends up how much confidence one had in one's own ability to produce. It seems likely that Tesla did not have enough confidence in his own abilities to refuse to work for Edison. It is true that one of the things Edison was much better at than Tesla was "blowing his own horn"! Perhaps because of that, and the fact that he was working under Edison, Tesla does not get as much credit as he should but he certainly is well known to physicists and engineers. As for "the key to unlimited and free energy", that's non-sense. Tesla, and Edison, were inventors and engineers, not physicist. Nothing Tesla was working on had anything to do with refuting or altering the laws of physics and "unlimited energy" would violate a basic law of physics. I hadn't read that Tesla had claimed to do all those things but they are certainly not true, especially "teleporting" anything.

Actually, an octopus does have a hard bone around its brain- but what SFN said is the best answer

No, it doe not show that h(g) is the inverse of f. You actually have to show two​ things- that h(g(f(x))= x and​ that f(h(g(x))= x.

You can ​approximate ​an integral by a Riemann sum. Is that what you mean by "replace"?

Yes, that is true. I think that you would need to support the mirror with some backing that will not change much with temperature. 35" is pretty large for an amateur telescope. What experience do you have with making smaller telescopes? It is also true that the cost of a telescope, mostly in the supporting structure, is roughly proportional to the cube of the diameter of the lens.

Rather than actually finding f(f(x)) you can do it this way: let y= f(x) so that f(f(x)) is f(y)= 4y- 5= 23. Solving that, 4y= 23+ 5= 28 so y= 28/4= 7. So now y= f(x) becomes 7= 4x- 5. Again, 4x= 7+ 5= 12, x= 12/4= 3. Check: f(3)= 4(3)- 5= 12- 5= 7 so that f(f(3))= 4(7)- 5= 28- 5= 23.

With samples of the man's DNA and the DNA of supposed parents, grandparents, etc. you can determine whether or not they share enough DNA to actually be "parents, grandparents, etc.". But the man's DNA alone will not "identify" his family tree. Perhaps you are referring to those "services" that use your DNA to tell you "where you are from"? Those do not determine your family tree- they simply identify parts of the world where that DNA is very common.

What do you mean by "type of media behavior"?

Assuming you are not including air resistance (which would make this problem far, far harder) the kinematic equations would be the usual [math]s= (a/2)t^2+ vt+ d[/math] where a is the acceleration vector, v is the initial velocity vector, and d is the initial position vector. Separating x (horizontal) and y (vertical) components and taking the initial speed to be "v" and the intial position to be d= (0, 0), we have [math]x= v cos(30)t=(\sqrt{3}/2)v t [/math] and [math]y= (-g/2)t^3+ v sin(30)= -4.9t^2+ (0.5)vt[/math] where v is the initial speed. Since the ball is to end up "20 meters away, the top edge is 5 meters above the throwing point", x= 20 and y= 5. Solve the two equations [math](\sqrt{3}/2)v t = 20[/math] and [math]-4.9t^2+ (0.5)vt= 5[/math] for t and v.

"In probability, it used to be something like: the maximum amount something could change based on a single event. However, now I can't find any evidence of that old definition." I'm pretty darn elderly and I have never seen such a definition. The probability definition of "variance" is the sum of the square of x times the probability x occurs, [math]\sum (x- \mu)^2P(x)[/math], where x varies over all possible outcomes (and, in the case of a continuous variable, the sum becomes an integral).

Well, what are Kepler's laws?

As far back as WWII there were gliders capable of carrying small tanks. Exactly how much weight are you taliking about? And how far do you want to carry it?

Combining bleach and ​ammonia ​will release chlorine gas which is a dangerous poison. Are you sure that wasn't what you read?

Your last statement, "When the two objects were far apart, their combined energy was less than the newly formed object has" is wrong. As Dr. P said, you need to include "potential energy".

What, exactly, do you mean by "divine"? For that matter what do you mean by "exists"? You start talking about whether the cat is "dead or alive" but then switch to "exists". I would say that once I put the cat in the box or saw someone put the cat in the box, I know it "exists" whether it is alive or dead.

Wolfram did not work because the primitive of this function is not an "elementary function" (so cannot be written in terms of trig functions, exponentals, logs, etc.).

Wonderful! Absolutely hilarious! (This was a joke, right?)

Mordred referred to the more common phrasing of this so-called "paradox"- the rigid rod, rather then your string- but the explanation is the same. Because information cannot be sent faster than the speed of light, there cannot be, under the assumptions of relativity, there cannot be a "rigid rod" or a "non-extensible" string. When you pull on a string, you stretch the string and it is the stretch that is transmitted along the string. Actually, the information is only transmitted at the speed of sound in the string or rod.

"Parts per million" isn't a good measure here. I assume you mean each "100 nanometer" piece to be a "part" of gold but what is a "part" of water? You would be better of calculating density in terms of grams of gold per gram of water.

What is preventing you from making one yourself? It couldn't take more than about 30 minutes.

Do you at least understand that "speed" has to be measured relative to something? Your first question is simply meaningless. As for your second question, there are still things such as heartbeat, breathing rate, etc. that give a rough measure of time.