In a square ABCD with side length 4 units, a point E is marked on side DA such that the length of DE is 3 units.

In the figure below, a circle R is tangent to side DA, side AB, and to segment CE.

Reason out and determine the exact value of the radius of circle R.

I'm developing a software library for working with 1d and 2d shapes, and one of the common operations I need is sampling a random point on a shape. For 1d shapes (line segments, Bezier curves, etc) there is a way that I find quite elegant:

let curve = ...some Bezier curve or line segment...;
curve.parametric(random())

Where curve.parametric(...) takes a value from 0 to 1 and returns the corresponding point on the curve, and random() produces a random value from 0 to 1. This form is useful not only for random sampling but for other operations as well, like finding the midpoint (just pass 0.5 in there).

But now I need similar functionality for 2d shapes, like concave polygons and ellipses. Is there a similar "parametric" form that would allow me to elegantly get a uniformly distributed point within the shape by passing in some random numbers, while also being useful for other geometric operations? Or is implementing a special getRandomPoint(...) function the only reasonable option here?

Thanks!

How would you prove this statement (highlighted in the image)? It's not clear that this statement is true whether you mean internal or external direct sum. It's also not immediately clear that this is necessarily infinite dimensional.

Unfortunately the author hasn't actually defined the notion of a module basis. Presumably it is essentially the same as a vector space basis. I can see how every vector field X in T(U) can uniquely be written as Xⁱ∂_xⁱ simply by considering its value at every point p, using the differentiability of X and the fact that ∂_xⁱ(p) is a basis of T_p(M).

So I'm a bit lost on this one becuase I was sick when we did this in school so I got a tutor but I cannot figure for the life of me what happened in this task

f(2x-3)=-6x+12

t=2x-3

2x=t-3/2

x=t/2 and 3/2

And then I should just add the t/2 and 3/2 in -6x+12

but the problem is I'm quite lost where did the 2x=t-3 come from?

i'm confused whether the total area is 2 square units or 2√2 square units. please help me out, a detailed explanation with the answer will be greatly appreciated

The question is pretty clear. It's pretty easy to find an example when the function is decreasing, but it seems far more complicated in reverse. I asked AI to help, because the question is far above my grade. Sadly, it could not construct such a function. I have barely any serious mathematical education, so I am not even sure how to proceed. Maybe there is no such function, but I could not fathom how to prove it.

Wikipedia says: In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof

And:

In mathematics, an analytic proof is a proof of a theorem in analysis that only makes use of methods from analysis, and that does not predominantly make use of algebraic or geometrical methods

Is there also a kind of proof theory that opposed to analytic proofs has algebraic proofs or something like that?

Hey there!
I'm looking for some help with calculating the fuel efficiency of my car. I've been trying to figure it out in google sheets, but I'm not quite sure I'm doing it correctly.

Each time I fill up the tank, I record a few parameters:

• Current mileage
• Amount of fuel added. (I always fill up the tank to the fullest. Tank capacity is 45 liters)
• Estimated range remaining before refueling (according to the car's onboard computer)
• Estimated range after refueling (also from the car computer)
• Cost of the fuel
• Date of the fill-up

I'd like to know how to properly use this data to calculate my car’s fuel efficiency for each period between last refill and current refill. Can you guide me through the process?

I have like tree pages trying different substitutions and still cannot solve this. I tried trigonometric subtitution, variable chage (u = denominator, u = x^a, ...). Can someone help me out or guide me in the right direction?

I hate typing! I really hope you can read my handwriting. I'll type the question anyway though... 4 people have 1/3 chance of saying the truth. A says, B denied that C claimed that D lied. Probability of D lieing?

Rolled 2 six sided die ~300 times without getting double sixes followed by rolling one six sided dice ~50 times without getting a six. What are the odds of that? I don't know how to calculate that.

consider any two natural numbers n and m

m < j < 2m where j is some prime number (Bertrand's postulate)
n < k < 2n where k is another prime number (Bertrand's postulate)

add them
m+n< j+k <2(m+n)

Clearly, j+k is even

And we can take any arbitrary numbers m and n so QED

failed maths in high school and never really paid attention is there an easy way to start again

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I was trying to understand what is going on in the set intersections (c) and (d) here?

I’m seeing this set notation for the first time so I’m trying to understand these.

Also was wondering how do you refer to these set intersections in words, when you say it out loud?

Upper expression is in phasor/complex/imaginary form.
Lower expression is supposedly the upper expression converted into time-form.

From my understanding you convert through Re{expression * e^jwt) and you'll get the time expression.
I however got -sin(wt-kR) as the last factor, which is not equivalent to the last factor of the proposed solution of my book, sin(wt + pi/2 -kR). It's not impossible there's an error in the solution but I doubt it.

Pythagoras says: c² = a² + b² Law of Cosines says: c² = a² + b² - 2ab·cos(θ)

The only difference is that last term: -2ab·cos(θ).

I get how both formulas are derived on their own, but I’m trying to understand why they’re so structurally similar, and why the correction term is specifically -2ab·cos(θ). More specifically:

If you take a right triangle and increase the 90° angle to something like 110°, keeping the 2 shorter sides’ lengths constant, why does the change in the opposite side’s squared length have to follow exactly the form of 2ab·cos(θ)? Why is that the specific correction needed? Is there any intuition, or is this merely a coincidence?

(I’m imagining keeping the base fixed, and rotating a line of length b θ degrees around one end of the line to form a circle. Thus the problem reduces to working out the distance from the circumference to some fixed point A, which is easily solved but doesn’t provide intuition for the original problem. Perhaps scalar product is useful? Not entirely sure.)

So the Biot-Savart law states that $\overrightarrow{B}=\frac{\mu_0I}{4\pi}\int_C\frac{d\vec{l}\times\hat{r}}{\left| \vec{r} \right|^2}$ and my question is what does that $d\vec{l}\times\hat{r}$ even mean, is it literally taking the dot product with a differential so $(dl_x,dl_y,dl_z)\times\hat{r}$ and then what is dl, it represents a small chunk of the curve so is it like the derivative of the curves times the diferential of the parameter that defines the curve? the concept of the law I get it but the maths not so much

A semicircle with diameter AB with center O is given. Any two points C and D are taken on it. Chords AD and BC intersect at point E. Let F be the projection of point E on the diameter AB. Prove that

a) The ray EF is the bisector of the angle CFD.

b) the center O is located on the circle circumscribed on the CFD triangle;

Hi all,

Hoping to get some opinions from you all on the use of a percentage value in an equation and ultimately the effects of that use in a final answer.

I am taking a statistics class where we are studying things like confidence intervals, hypothesis testing, etc., and a question came up that was slightly different because it involved values given to me in a percentage form, not as a plain decimal value. Now my professor does not want her test questions posted in places, so I am going to make up some numbers and give you the important factors.

The formula for the lower confidence interval, L, is

L = (n-1) s² / chi²

where n is the number of samples, s is the sample standard deviation, and chi² is a test statistic for the problem (doesn’t really matter for this question, but just putting it out there).

So lets say we are given n = 13, chi² = 20, and in this instance I tell you that s = 2.1%.

I ask you what is L to four decimal places?  How do you compute this?

I compute:

L = (13-1) * (.021)² / 20 = .0002646 (round to .0003)

The professor computes:

L = (13-1) * (2.1)² / 20 = 2.6460

Here I think there is an implication that this answer is in percent form, but that was not specifically stated by the problem question.

Now I contend that my answer is right, because all I did was take a percentage value and divide by 100, and I contend that 2.1% = 0.021 so I can make that substitution with no issues.

However,  I don’t think our answers are equivalent, even if you account for the fact that maybe you wanted your final answer as a percentage, because my final answer is still .02646% if I express it as a percentage, which is still off by a factor of 100 from the professors answer.

Are we in agreement here that my answer is technically correct because I got rid of the % sign immediately, and the professor’s is technically wrong because by squaring the percent value, they are essentially calculating %^2, or 1/10,000, which would certainly not be something that you would want to do in this type of problem.

Thoughts on the discrepancy?

Exercise 20. I was train my proving skills, but something goes wrong. Can you give me advice or idea how to prove that? I was thinking about it alot, but I really can't see how. I only know that I need to use a contradiction. But where I can find it?

So far, no one in my family can figure out how to solve this question. I assume it's from a math textbook but I don't know which one. We can't seem to find the relationship between the length and the number of cubes. My brother says the unit is number patterns but we can't seem to find one. Multiple people have already spend over an hour trying to figure this out. Are we stupid or is the question inherently faulty? Thanks in advance for the help.

I had a hard time putting this question into words but hopefully I can explain it with some examples.

Let's say you scored 50 out of 60 on a test and then the teacher decided to make the test out of 55 instead of 60,

Original score - 50/60 = 83.33%

Score after grading adjustment - 50/55 = 90.9%

Change in % = 7.57%

Now lets say you scored 30 out of 60 on the same test,

Original score - 30/60 = 50%

Score after grading adjustment - 30/55 = 54.54%

Change in % = 4.54%

I first thought would be that the % change would be the same regardless of the numerator. I can't wrap my head around why it isn't a constant change. Please explain in simple terms for a simple human (me) if possible!

The theorem's somewhat ^§ explicated in

Turán’s theorem: variations and generalizations

¡¡ may download without prompting – PDF document – 455·7㎅ !!

by

Benny Sudakov ,

in the sections Local Density, Large Subsets, Triangle-Free Graphs & Sparse Halves ... the sections that have the figures in the frontispiece in them.

§ That's the problem: only somewhat !

(BtW: this is a repost: there was something a tad 'amiss' with the link to the paper in my first posting of it. Don't know whether anyone noticed: I hope not!

😁

This time I've put the link to the original source in, even-though it's a tad more cumbersome.)

It's a recurring problem with PDFs of Power-Point presentations: they're meant to be used in-conjunction with lecturing in-person, really. But it's really tantalising ! ... in the sections Local Density, Large Subsets, Triangle-Free Graphs & Sparse Halves there seems to be being explicated an interesting looking variation on Turán's theorem concerned with, rather than the whole graph, the induced subgraphs thereof having vertex set of size αN , where N is the size of the vertex set of the graph under-consideration & α is some constant in (0,1) . But it's not thoroughly explicit about what it's getting@, and the 'reference trail' seems to be elusive. For instance one thing it seems to be saying is that if α is not-too-much <1 then the Turán graph remains the extremal graph ... but that if it decreases below a certain point then there's a 'phase change' entailing its not being anymore the extremal graph. If I'm correct in that interpretation then that would be truly fascinating behaviour! ... but I'm finding it impossible to find that wherewithal I can confirm it.

So I wonder whether anyone's familiar with this variation on Turán's theorem in such degree that they can explicate it themself or supply a signpost to the references that have so-far evad me.

I have a question asking which point IS NOT AN INFLECTION POINT, the answer is “f” which I can understand, however I’m wondering why the answer is not “i” either? That point is a cusp so I thought they could not be points of infection? To make it more confusing there is a question asking where f’’(x) = DNE in which the answers are a,g, i, and K. How can “i” be an POI but also does not exist on f”(x)? HELP