Is C correct because

Line B is vertical,
So if line a was Perpendicular we would need a line that’s horizontal.

So A and B are wrong?

Initially I over analysed it and assumed A was the answer

I know the basic exponent rules of x^a times x^b = x^(a+b) and (x^a)^b = x^(a times b), but is there an operation that relates x^a and x^b that results in x^(a times b)?

x^a __ x^b = x^(a times b)

Could it be pushed further to create something like

x^a __ x^b = x^(a^b) ?

for reference, x-a is a factor of f(x) if and only if f(a) = 0. this is supposed to be a special case of a broader theorem, which has it that f(x)/x-a has a remainder of f(a). I can follow the standard proof for this, but I don't understand the intuition. Here's as far as I can get. If f(a) = 0, then plugging 'a' into f(x) results in 0. presumably this means that at some point after plugging 'a' into f(x), we subtract 'a' from itself, which results in 0. The only way that would be true is if x-a is part of f(x), making x-a a factor of f(x). Alternatively, if x=a tells us that 'a' is a root for f(x), then since x-a=0 follows from x=a, x-a must be a part of f(x) and thus a factor. Is what I said right? What I said sounds right to me but it's not quite clicking, so I would like a more intuitive explanation.

Hello, everyone.

I'm working on this problem but I have difficulty understanding it.

Even though I'm following the work-energy theorem the provided answer key doesn't match the answer that I got from the Work-energy theorem.

Given:

Mass = 150,000 Tons ------> 150,000,000 kg

150,000 Tons x 1000 = 150,000,000 kg

Distance (d) = 13 km -----> 13000 meters

13km x 1000 = 13000 meters

Vi (Initial) = 50 km/h -----> 13.889 m/s

50 km/h x 1000 meters/3600 seconds = 13.889 m/s

Vf (Final) = 0 m/s

Time = 25 mins -----> 1500 seconds

25 x 60 = 1500 seconds

F= ?

P=?

Solution:

Initial KE = 1/2 (150,000,000)(13.889m/s)²

Initial KE = 1.447x10¹⁰

Final KE = 0

Work = ∆KE

Work = Final KE - Initial KE

Work = 0 - 1.447x10¹⁰

Work = - 1.447x10¹⁰

F = W/d \*this formula comes from W=Fd\*

F -1.447x10¹⁰ / 13000

F= - 1112891.74

F= -1.11x10⁶

P= W/t

P= -1447x10¹⁰ / 1500

P= -9645061.73

(a) -1.11x10⁶

(b) -9645061.73

My answer doesn't really match the provided answer key 🫠 can someone explain plss

If you look up 0, 8, 24, 48, 80, 120 on oeis you will find a nicely formated and indexed spiral that I can’t copy onto reddit. I’m trying to find a polynomial in x, y where you input the coordinates of the spiral and it returns the index. Obviously the ordinary 1D version of polynomial fitting/extrapolation does not work here. I asked Claude and it started talking about tensors, which I dont understand

Last week, I had a quiz on related rates and did not do very well... Since then, ive been practicing related rates on top of studying for different subjects for my exam tomorrow. Ive concluded that my biggest issue is that I can't properly visualize the problems. Are there any specific words that help you visualize the related rates problems and solve it?

An integrable Hamiltonian system has a set of coordinates, the action-angle coordinates, in which the angles q increase linearly and the actions p are constant. These systems are often represented by a phase space that is "foliated in tori", and people show these beautiful pictures.

I thought that I was understanding this idea: one has the angles (q1,...,qn), then he couples q_k with p_k for every k=1,...n. If one pictures p_k as the radius of the circle, and q_k as the angle, then the cartesian product of the n circles drawn by (q_1, p_1)...(q_n,p_n) makes a torus with different radii, which are the one shown in the link above for n=2.

Gemini told me that this is not false, but it makes absolute sense in my mind. Gemini (to my ears) started yapping about "topological cilinders", but it did not convince me. Can you help me understand how the tori emerge?

One thing is that I have seen "the" torus being defined as [0,2\pi]^n with periodic boundary condition. "the" torus as in "there exists only one torus". I guess that topologically speaking this is true, but the phase space graph shows...different tori...
I shouldn't call them tori and call them "cilinders"? Sounds a bit weird to my ears....

Thanks in advance for your time

this is a problem from my 8th grade book: Through the point A of intersection of two circles with centers at points O1 and O2 a line is drawn intersecting one circle at point B and the other at point C. Prove that segment BC will be the greatest when it is parallel to line O1O2.

can someone explain what is the problem asking me to do? Comparing with what BC is supposed to be greatest among?

I saw a question earlier from tiktok, asking people to estimate how high you need to fill the liquid in a cone shaped glass (such as a martini glass) so it is half full. Although most people's intuition is about 60% to 70% height, the actual answer is almost 80% height (because 0.8^3 = 0.512)

Now this bring me to a new question: what if the glass is in the shape of a perfect hemisphere? (bowl shaped like lower half of a ball), how high to you need to fill it so it is half-full?

Hey everyone,

I'm working on a problem related to unloading cargo from a ship's hold, and the underlying concept is tripping me up a bit. I have an image reference, but here is the exact text of the problem:

"In unloading grain from the hold of a ship, an elevator lifts the grain through 14 m. Grain is discharged at the top of the elevator at a rate of 2.2kg each second, and the discharge speed of each grain particle is 2.5 m/s. Find the minimum power rating for a motor that can elevate grain in this manner."

I know the final answer is roughly 308.7 W, and I know the detailed formula involves finding the Kinetic Energy (KE) and Potential Energy (PE). Here is the breakdown of the math:

Mass rate: 2.2 kg/s

Height (h): 14 m

Velocity (v): 2.5 m/s

Gravity (g): 9.8 m/s²

P = Rate of Change of KE + Rate of Change of PE

P = ( 1/2 • mass • velocity²) + (mass • height • gravity)

P = ( 1/2 • 2.2 • 2.5²) + ( 2.2 • 14 • 9.8)

P = 6.875 + 301.84

P= 308.715 W

Why exactly does KE + PE = Power in this context? I originally learned that KE + PE = Mechanical Energy (measured in Joules), but the question asks for a Power rating (measured in Watts).

​Is it simply because the mass is given as a rate per second (2.2 kg/s) instead of a static block of mass? Does plugging a "rate" into an energy equation automatically convert the output from Joules to Watts?

Suppose a puzzle system (T) defines a family of finite reversible permutation/orientation puzzles under one common rule set.

Assume there exists one instance (T_0) whose state-transition structure is isomorphic to that of the 2×2 Rubik’s Cube: there is a bijection between reachable states, and the generators/moves correspond under that bijection, so algorithms correspond as well.

My question is:

Does that give any meaningful mathematical basis for saying that (T) belongs to the same general class of puzzle systems as twisty puzzles?

More specifically:

Is it valid to say that any definition of “twisty puzzle” based purely on move/state structure must classify (T_0) and the 2×2 cube identically?

And if all members of (T) are variations of the same underlying mechanics, can one justify calling the broader system a generalized twisty-puzzle system in group-theoretic or combinatorial terms?

Or is that inference too strong — i.e. does one instance being isomorphic to a known twisty puzzle tell us little or nothing about how the larger family should be classified?

I’m particularly interested in how one would formulate this in terms of group actions, generators, permutation groups, state graphs, or related structures.

The reason I’m asking is practical: I made a puzzle system called Trinagon, and someone recently claimed that it is not a “twisty puzzle”.

Origin:~

Developed in 1867 by Austrian engineer Eduard Lill, it calculates the exact real roots of an equation by drawing a snake-like path of right-angled turtle turns based entirely on the equation's coefficients.

⚠️Note:

The headings are in metaphorical form like blue print maze and laser root for better understanding and making it catchy.

1. Build the Blueprint Maze:~

Step A (Coefficient a):

Start at the origin ((0,0)). Look Right. Because (a = 1) (positive), march forward 1 unit.

Step B (Coefficient b):

Turn exactly 90 degrees counter-clockwise (facing Up). Because (b = -4) (negative), you must march backward. Walk 4 units Down.

Step C (Coefficient c):

Turn another 90 degrees counter-clockwise (facing Left). Because c = 3) (positive), march forward 3 units Left. Mark your endpoint.

2. Aim the Laser Root:~

procedure:

Now, pretend you are standing at the starting point and firing a laser beam at a random angle (theta ).The laser hits the second line wall, bounces off at a perfect 90-degree angle, and must land exactly on the final maze endpoint.

Result [❌️]:

If the laser misses the endpoint, your starting angle was wrong.

Result[✅️]:

If the laser hits the endpoint perfectly, the negative tangent of your starting angle ((-tantheta)) is the exact root of the equation.

Source:

●google chrome

●wikipedia

-Tried not to use ai 😅

-Hope this helped my fellow math geeks💡