First, I need to write a dynamic equation that describes how the tank's water level varies over time as a function of Fin, Fout, H, Lt, and Lb. Then, I have to linearize it around an adequate operating point. I'm stuck, and I can't get an answer. Could someone help me? Consider the tank shaped as a truncated pyramid. The height of the truncated pyramid is H,
assume a square base with length Lb on the bottom and Lt at the top. Assume that Fin and Fout are the volumetric flowrates of water in
and out of the tank, respectively

There are different natural numbers written on a board with the sum of 2024. Each of them, except the smallest one, is a multiple of the sum of all the smaller numbers written. How many numbers at most can be on the board? Hope you guys could help me, thank ya :)

I've calculated 63 ( 1+2+3+...+64 = 2080 so we have to subtract 56 )

I asked chatGPT for recommendations and it gave me "Introduction to set Theory by Jech" and "Elementary Number theory by Burton". I started them and Im not super impressed. Are those good? any better ones? I am somewhat of a beginner but I did a few textbooks on proofs that had introductions to both subjects.

So, I saw a simple linear system used as an example. I understand the Gauss method but what I noticed was, after finding the value of y, "x" will have different values depending on which equation I substitute. I have never seen any book mentioning this detail, thats why I want to confirm. The equation and its transformations are below:

2x+3y=6
x−y=1/2

After applying Gauss elimination, we will have

1 -1 12

0 1 1

Thus, we can say y=1. But, the book uses it to substitute the "y" value on the original equation, like 2x+3y=6. What I dont understand is why cant we substitute in the upper row ( 1 -1 12), so it would be x-y=12, which would give x=13? Thats a completely different result than the given answer, that is x=1/3 and y=1.

But if Gauss elimination doesnt change the values of the variables, how come doing the substitution in the upper rows will give me wrong results? Can someone please explain this?

If I have "n" number of things and I am splitting them into "a" groups of "n/a" things, what is the probability of one specific thing being in a group with another specific thing (or multiple specific things)?

For instance, a class of 33 is split into 11 groups of 3. What is the probability of being paired up with your two best friends? (If you only have two.)

As an extension to this question, how do I calculate this of I have "k" friends, or when I want to be in a group with at least "m" friends. (or both at once)?

For reference, I have done the grade 12 permutations and combinations unit so I won't be confused if you say 3C2.

i'm currently in 9th grade (brazil) and am trying to learn calculus by myself, given that it is not a highschool subject in my country. I have picked some of my dad's books (he's an engineer), and went through some pre calc stuff on khan academy. I believe the only stuff that I still don't know is vectors, analytical geometry and conic sections. I was wondering if anyone could give some basic pre-requisites that I might be missing on.

Let’s say we use z test statistics (population standard deviation is known and the sample size is large enough). If we have a left tail, whether the test score is positive or negative, we can directly use cumulative probability( found from z table) as a p-value. If we have a right tail, we can directly use cumulative probability as a p-value if test statistics is negative. But if test statistics is a positive number, we will subtract cumulative probability from 1 to find p-value. If we have two tails and if test score is a positive number we will subtract from 1 and multiply to 2 the cumulative probability to find p-value. But if test score is a negative number, we will just multiply cumulative probabilities by 2 to find p-value

In this homework set we're using linear transformations to create an isometric view where the plane α is the screen and the angle θ is the angle at which the screen is viewing the scene.

In the previous problem, we calculated θ by using the transformation matrix to get a vector in. The apparent x direction and multiply it by a purely horizontal vector representing the apparent y direction, but I'm struggling to understand the relationship between k and θ since it gets multiplied with the rest of the vector when we're getting the apparent x vector

https://drive.google.com/file/d/1sGucUpvpbI6BcKtxE9IDWKacyWe9TuN-/view?

I urgently need help for my quiz question.

Say that V is a vector space. And dimV = n. For {alpha1, alpha2, …, alphan} system

1 - If it’s linearly independent, then it satisfies the span axiom.

2 - If it satisfies the span axiom, then it is linearly independent.

I need to prove this theorem. We’re going to have a quiz tomorrow and this will bee the question. All of my friends and I have been thinking about a solid proof for days and we couldn’t find anything. Normally I’d never post but I really need help lmao

Normally a system doesn’t have to be linearly independent to span a vector space but the trick/problem here lies in the dimension of the space, I think.

In general form if possible with steps :)

I have a bit of confusion, for a graph to have eulerian path, it must have atmost or exactly 2 vertices with odd degree?
if there are zero vertices with odd degree, is it eulerian path or circuit?
same again if there is exactly one vertex with odd degree, is it eulerian path or circuit?
and etc, if atmost 2 or more than two...

thanks if you can answer them all.

Hello, i apologize if this is a stupid question but i’ve watched every youtube video i can find and still can’t figure this out

to start i struggle immensely with factoring and i finally found a way that makes the most sense to me. the method where you find 2 numbers that multiply to the third term and add to the second. i’m just factoring this trinomial x^2 -2x-3 and i don’t understand why the answer is (-1, 3) and not (-3,1) the textbook says (-1, 3) is the correct solution and i have no clue why. help would be immensely appreciated, thank you.

Need help on this question

The question reads “In a volatile housing market, the overall value of a home can be modeled by

V(x) = 415x^2 - 4600x + 200,000

Where V represents the value of a home each year after 2020.

Part A: Find the vertex of V(x). Show all work. Part B: interpret what the vertex means in terms of the value of the home.

Given is an acute triangle ABC in which AB < AC. Let us denote the circle circumscribed by S in triangle ABC. The points M and N are the centers of the longer arc BC and the shorter arc BC of the circle S, respectively. Points X != M ¡ Y != M lies on the line AM and satisfies the equality BX = BM = CM = CY. Point E is the orthographic projection of point B on the line AC, and point F is the orthographic projection of point C on the line AB. prove that the angle fnx equals the angle yne.

I did the drawing and well started thinkg about the arcs abd why would bm=cm what can it give to us... Any help appreciated.

From my understanding 1(1/x)/2 = (1/2) (1/x), so there should only be an x in the beginning of the denominator. Have the link in my comment.

Long story short: I am 29 and thinking about joining the military. To do so you must take a test called the ASVAB so basic mathematics and arithmetic reasoning.

I’ve been polishing up on my skills for a little over a week and most of it is coming back to me ( haven’t used these since high school so I was definitely rusty)

Arithmetic reasoning is killing me. I do have dyslexia ( you can still join with this) and I believe it’s just causing my brain to not be able to compute what I’m seeing in front of me. Obviously with any math problem once I know the equation I can try and solve it. I just can’t seem to notice what I need to impute to solve for the question. I’ve watched YouTube videos and even have a practice book for the asvab.

Do any math wiz’s out there have any insight or tricks to help me with this? I know d=rt & I=prt can be distinguished pretty easily, but I guess word problems that don’t use these are throwing me off mostly.

when P <=>Q, why does this strictly mean that P Q must be true for P to also be true , and vice versa, well indeed each implies the other, but why would that indicate that at one time either both or none are true?

is there a way to rigorously define something like a>b? I was thinking of

if a>b, then there exists c > 0 st a=b+c

does that work? it is a bit of circular reasoning cuz c >0 itself is also an inequality, but if we can somehow just work around with this intuitively, would it apply?

maybe we can use that to prove other inequality rules like why multiplying by a negative number flip the sign, etc

When you put in a graphing calculator like desmos or geogebra f(x) = x times cos(x) or f(x) = x times sin(x)
the waves of the graph instead of going up and down at a fixed rate, the waves go up and up and up and don't stop increasing. I've tried searching informative videos and asking people but I don't really know the name of this so I haven't really gotten any results. I would appreciate any explanation, thank you!
https://imgur.com/a/jY2a1Qf <--- image of graphing

Okay, so let's say I have 5 people starting with bob at 18, Cody at 12, Lisa at 10, Emily at 7, and Nora at 2.

Every turn bob subtracts 6, Cody subtracts 2, Lisa subtracts 8, Emily subtracts 4, and Nora subtracts 5. When they reach they get to move, and start over from 20 , subtracting the remainder. (iE Nora is at 2, her subtraction is 5 so she ends up being allowed to move and ends up at 17)

What I need is a way to quickly keep track of who gets to 0/20 first/next and keep the game moving.

I needs to include a theorem and diagrams so any ideas of what to research would be helpful.

These two terms are used seemingly interchangeably and the Wiki for the Poisson process even redirects to the Poisson point process. Yet I still see people go out of their way to include the "point" in the Poisson point process so I am starting to think one is a generalization of the other. Is there any difference?

They both compute the area of a function along a curve traced out by another function, so geometrically they appear to be identical. Is there any difference between them?

Going through a College Algebra textbook, and got presented with this problem:

Factor 2x^(3)−12x^(2)+6x

I got 2x(x^(2)-6x+3) ; and I put this solution into several other math solvers, and that was the correct solution, however in the textbook I am going through, the solution was said to be:

12x(x^(2)-6x+3)

Which is correct? The textbook I am going through is the UWCABook - College Algrebra Version 3.

Hi All. I struggle with simplifying terms containing bra-ket notation. For example, how can I simplify (|01><00|+|11><10|)(|0><0|+I/2) where I is the identity. Any insight will be highly appreciated. Thanks in advance.

Hey all. Please help me understand something. It seems like there are so many ways to go about OoO, even though there really shouldn't be.

If we had ONE system we agreed on, and ONE system that was taught consistently, I think it would be extremely helpful. CLEARLY, because people are always getting different answers, (usually '1' and some other non-descript number). IF THE ACCROYNMS *REALLY* WORKED - THERE WOULD BE ZERO AMBIGUITITY.

My last math class was something I call, "if you didn't remember everything from grade school, we're getting it back NOW." College MAT 120/ applied mathematics. We went from literal Adding to Trig, and I really loved it! I got an A, and ALWAYS used PEMDAS. As you'll see I forgot some stuff already... It was a long time ago.

Here is the problem in question. I'm not going to say what I think it is, but I'll say how I could see the confusion. Tbh - I'm not 100% on my answer myself.

6 / 2 (1+2) = ?

Here is how I can see both "possible answers" of 1 and 9, using PEMDAS for both.

(answer of 1):

6 / 2 (1+2)

6 / 2 (3)

*in this instance, we're doing multiplication first, because M comes before D in PEMDAS

6 / 6 = 1

(answer of 9):

6 / 2 (1+2)

6 / 2 (3)

*in this instance, we're doing division first, because in "similar" operations (multiply/divide, add/subtract), you go from Left-to-Right in a problem like you're reading a book.

3 (3) = 9

Now here's the thing.... I don't even know how useful this post is, because as evidenced by wherever this problem appears on the internet, there will always be conflict. But I have to ask... which one is correct? I LITERALLY remember both rules about PEMDAS existing at the same time. I have a feeling they can't - because clearly they create dilemmas like this one. I have a hunch, but I'm not sure...

Trying to teach myself a little bit of maths and am doing the questions for the first exercise. I can't find out how to get to the solution given for question 8:

f(x)= x² + 2x + 1 Find and simplify an expression for f(x+δx) - f(x)

The solution in the back of the book is:

2x.δx + 2δx + (δx)²

Could someone please help and include steps for working this out?

1.an+2-6an+9an=3(2^n)+7(3^n)

2.an=4an-1-4an-2+(n+2)2^n

P.S I would really appreciate books or other resources I can use to learn these

The cubic root of(x-2)= the square root of (x-2)

I tried solving this problem by giving the exponent 6 to both sides, then factoring it out and by doing this I get x^3-x^2+4x-12=0 I divide this by (x-2) to then get x^2+x+6 * (x-2) =0 Solving it I get that x=2 However when I put the equation into a calculator it shows that x also equals 3, why is that?

Have someone use smartyme app for math . Need some feedback