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Since the set is composed of 18 consecutive integers, none of which is divisible by 19, the set must be equivalent to the set $$\{1,2,\cdots,18\}$$ when reduced modulo 19. If this set can be partitioned into two subsets with the same product, the product of the elements must be a square. By Wilson's theorem, the produc... | {
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By the binomial theorem we have that, $$(a + b)^p = \sum\limits^p_{k=0} \binom{p}{k} a^k b^{p - k}$$
By Problem 3, we know that $$1 \leq k < p \Rightarrow \binom{p}{k} \equiv 0 \pmod p$$, and so all of the summands except for the first and last ones are zeroed. We are then left with,
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# If the balloon is subjected to a net uplift 5
If the balloon is subjected to a net uplift force of F = 800 N, determine the tension developed in ropes AB, AC, AD.
Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.
#### Solution:
... | {
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$u_{AC}\,=\,\left(\dfrac{2}{7}i+\dfrac{-3}{7}j-\dfrac{6}{7}k\right)$
$u_{AD}\,=\,\left(0i+\dfrac{2.5}{6.5}j-\dfrac{6}{6.5}k\right)$
The unit vector is each corresponding unit of the position vector divided by the magnitude of the position vector. If the position vector was $r\,=\,ai+bj+ck$, then unit vector, $u\,=\,\... | {
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## 5 thoughts on “If the balloon is subjected to a net uplift”
• questionsolutions Post author
Where is it wrong? Just saying solution is wrong doesn’t help anyone, please point out what value is wrong, it’s easy to miss when checking so it would be very helpful if you can point out the error. I checked through and n... | {
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### Puzzle of putting numbers 1-9 in 3x3 Grid to add up to 15
• In a 3x3 grid, I'd have to put numbers from 1 to 9 in a manner so that respective row, column and diagonal add up to 15.
I have only been able to come up with one solution:
$$\begin{array}{ccc} 6 & 1 & 8 \\ 7 & 5 & 3 \\ 2 & 9 & 4 \end{array}$$
Through ... | {
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*Note: So as you are in the middle of the top row on the first move you want to place the next number in the next column of the row above. The row above does not exist so move to the last row of the square in the same column. If you were in the last column you would move to the first column. If you look at the example ... | {
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# One Team, Two Teams, My Team, Your Team
#### (A new question of the week)
Counting ways to select teams can be simple, or quite complex. Here we’ll look at a few tricky examples.
The question came in early April:
Dear Ask Dr Math,
I hope you can help me with the question below:
Four friends attend different sch... | {
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Here’s one hint that may help: Choosing two teams of 6 is identical to choosing one team of 6, because everyone not chosen is on the other team. Similarly, in the first question, choosing a team of 6 is identical to choosing two schools. Do you see why? This sort of thinking (thinking about how the choice might be made... | {
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But you don’t want to double that to get the final answer. What I think you mean is that either of the two teams chosen might be the first, so you double the number. But it actually works the other way, since (as discussed in the post I referred to, near the end) choosing two teams ignores order; there is no first or s... | {
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Here are the three possible pairings:
For a larger problem, we couldn’t have just listed choices to check our count; this is one reason I often check my reasoning by trying a smaller version of the same problem. Imagine if there were 50 schools competing …
## Problem 2: friends on the same team
Zack replied:
Thanks... | {
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What do you think about my solutions and answers?
### A subtractive method
My response continued:
This problem is much more complicated, and this is a good attempt, though flawed. You’ve done it a different way than I did, and got a different answer. So let’s see who’s right! We won’t want to check by listing this t... | {
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So his plan is to count these and subtract them. But there’s another problem:
Second, you will be overcounting, because you will get the same result if you take, say, person A1 among the first four and then pick A2 as one of the extra two, or vice versa. This is a common flaw in this sort of thinking, and sometimes ve... | {
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### An additive method (not)
Clearly you are capable of thinking through these questions, so I’m not going to show you my answer yet. I will tell you the basics of my approach, however; I don’t know how much it will take to rescue your current approach, and this may help if you either give up on that or want to try a ... | {
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Zack replied:
Do I calculate AABBCD this way?
(4C2/2) × (3C2 × 3C2) × ( 3C1 × 3C1) = 243
I select 2 students from 2 schools (3C2 × 3C2). Since there are 4 schools I multiplied by (4C2). Since my approach will cause duplicates, then I need to divide by 2. I further multiplied by (3C1 × 3C1) to fill up another 2 spots... | {
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To that, I replied:
This is a different subtraction than you tried before; I haven’t tried to see whether that could be made to work. This time you are starting from all possible pairs of teams and trying to subtract every arrangement that doesn’t have at least one from each school on each team. I like that you are co... | {
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Finally, since the teams are not distinguishable, we divide by 2, and get the same answer we got before, $$243$$.
We’ve seen three different ways to get the same answer, which is not unusual in combinatorics!
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {
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# How can I get maximum number of vertices if I already know edges
If I already know edges how can I get the maximum number of vertices?
Question: There is a graph that has $$36$$ edges, and where every vertex has degree at least $$5$$. What is the maximum number of vertices this graph could have?
I think the sum of... | {
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# Solving complex equation
How do i further solve the following complex equation:
$$z\cdot \bar{z} + z + \bar{z} + i\cdot z - \overline{i \cdot z} = 9 + 4i$$ $$a^{2} - b^{2} + 2a - 2b = 9 + 4i$$
How do i solve from here on ?
-
Do you mean $z\bar{z}+z+\bar{z}+iz-\overline{iz}=9+4i$? – André Nicolas Aug 29 '12 at 14... | {
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# Exponential functions problem
## Homework Statement
In the 2 following problems they use the term in the brackets differently, in one case its a percentage and in the other case i have no idea where they get the number from, this is what i would like to find out
A cell loses 2% of its charge every day
C is total c... | {
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For the 2nd equation,
$$P(t) = 24(1.014)^t = 24(1 + 0.014)^t$$
... so the rate of growth is 1.4% per year since 1981. This information should have been given in the problem somewhere.
I usually think of exponential growth/decay models in terms of this equation:
$$P(t) = P_0 (1 + r)^t$$
r is the rate of growth/decay. r... | {
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# 10.4 OFSA Solutions
OFSA SOLUTIONS for Parametrics
This page contains peer generated solutions and error explanations to OFSA questions. As you read or view the solutions, be critical: check for accuracy, but also for more efficient solution strategies. If you have a better method or different idea/answer, post a d... | {
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Therefore, the correct answer is c.
*a correct, full description of the motion:
The particle begins at (0,5) and continues on the path defined by
$$\frac{x^{2}}{9}+\frac{y^{2}}{25}=1$$
in a clockwise direction and continues indefinitely.*
Error Explanation 1
a. the student found the correct graph, but assumed that th... | {
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$$y=x+30=90-x$$
$$2x=120$$
$$x=60$$
d. correct!
Question 3
Given:
$$x=\frac{2}{t-3}$$
and
$$y=\frac{1}{t+5}$$
eliminate the parameter, and choose the correct graph.
a.
b.
c.
d.
Solution 3
1) To eliminate the parameter, the most efficient way is to isolate the t in the x-t equation.
$$x=\frac{2}{t-3}$$
$... | {
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Solution 4
SIMULTANEOUS SOLUTION/COLLISION OF BALL AND FOOT
1. let X=X
2t+1=3t-10
t=11
2. let Y=Y
t-4=-t+3
t=
$$\frac{7}{2}$$
3. t:
$$11\neq \frac{7}{2}$$
NON-SIMULTANEOUS SOLUTION
Since the two paths intersect at different times, there is no collision and the paths cross at a non-silmultaneous solution. But where?
1. ... | {
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$$y=3\sqrt{-4+4}-4$$
$$y=-4$$
Error Explanation 5
a. Because the parameter 't' is inside a square root, it is impossible that t can extend beyond -4. Under the square root, the number cannot be negative, and since t is not squared, a negative will stay negative.
b.When the parameter ''t' has been limited to t:[-4,∞), ... | {
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Error Explanation 6
a. While on Parametric mode, this is the negative vertical height the calculator's table reads when the maximum horizontal distance has been reached.
b. The process was done in reverse, rather than plugging 0 into the y equation and 't' into the x equation, 0 was plugged into the x equation and the ... | {
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Solution 7
$\\ \text{1) let } x_1 = x_2 \\ 10\sin(6t) = 10\cos(6t) \\ \sin(6t) = \cos(6t) \\ \tan(6t) = 1 \\ \\ \text{2) let } y_1 = y_2 \\ 4\cos(4t) = 3\cos(4t) \\ cos(4t) = 0 \\ \\ \text{3) list possible t values for both x and y in the restriction of t.} \\ \tan(6t) = 1 \\ 6t \in \frac{\pi}{4} + \pi k \\ t \in \frac... | {
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Solution 8
$\\ \text{1) Let x1 = x2} \\ t + 17 = 3t - 1 \\ 18 = 2t \\ t = 9 \\ \\ \text{2) Let y1 = y2} \\ 5t^2 - 2t = 36t + 63 \\ 5t^2 - 38t - 63 = 0 \\ (t - 9)(5t + 7) = 0 \\ t = 9, \; \frac{-7}{5} \\ \\ \text{3) Find t values that solve both the x equations and the y equations} \\ \box{t = 9} \\ \\ \text{4) Find the... | {
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Yes, the man will step on the bug at (23, 11) when t = 9 seconds.
Error Explanation 10
a) Incorrect...There is a simultaneous solution
• I decided this could be a mistake a person can make if they have minor calculation errors
b) Incorrect...The simultaneous solution does not occur at (-13, -13)
• This mistake would b... | {
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Error Explanation 11
a) Correct description of motion of the object, with correct direction and specificity of the turning points
b) Incorrect...Since x = 7 (and is therefore a constant), the object must travel vertically
• The mistake for this would be that the person saw the t-x graph and mistakenly thought that the ... | {
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So the correct answer is $\small \bg_white \fn_cm 4\sqrt{13}$
Error Explanation 12
a) Incorrect...
• The mistake would be that the person forgot to multiply the distance by 2 because they figure they found the distance traveled by finding the distance between the two points
b) Incorrect...
• The person is thinking of ... | {
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Error Explanation 13
A) Correct! From t: [-5,-2], the particle is increasing in y-coordinates to 0.
B) Correct! From t: [0,5], the particle is increasing in y-coordinates to 149.
C) Correct! From t: [-2,0], the particle is decreasing in y-coordinates to -4.
D) Error. Student must have seen first increase in y-coord fro... | {
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x= 89cos63(4.956)
x=200.257 ft
The field is measured in yards, so to convert this to the correct unit we divide by 3, as a yard consists of 3 feet. We arrive at the conclusion that the ball travels approximately 66.75 yards.
4) The seperation between the lines of the field is 10 yards. Since the ball travels 66.75 ya... | {
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Solution explanation:
1) First write an equation that models the x behavior of the particle in relation to time. This will be your t-x graph.
• We see that the particle at t=0, begins its motion at x=0. As time progresses it moves from zero to its max value, back to zero, to its min value, then back to zero. This oscil... | {
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Errors Explained:
a) Incorrect period. May have seen t=12 being the largest t value and assumed that was the total period but remember, it must continue back to t=0.
b) May not have realized that the y movement is a NEGATIVE cosine curve as it begins from the min vs. the max.
d) The x and y are switched...read carefull... | {
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The graph of this particle is this:
(*The horizontal axis is defined by t and the vertical by y.)
• Displacement: At t=0, the particle begins at -5. At t=5, the particle ends at 0. Displacement is 5, because it is the difference in beginning and end positions.
• Distance: At t=0 the particle begins at -5. During this... | {
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Solution 20
• graphing both parametric equations yields these domains and ranges:
• x: [-3,3], t: all real numbers
• y: [-1,1], t: all real numbers
• note the trends of each graph to figure out the orientation!
• as t increases, x decreases to -3 then increases to 3 then decreases again
• as t increases, y decreases t... | {
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$x \neq \pm \sqrt3 \\$
• this is because the simultaneous solution lies there and we are only looking for the non-simultaneous solution!
• therefore, the correct answer is choice C) (3.32, 25)
Error Explanation 21
A) Error. This is the simultaneous solution. Correct answer if question was asking for the simultaneous ... | {
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# Math Help - Solution of Cubic
1. ## Solution of Cubic
Can someone help me find the exact solutions to the equation
$x^3 + 2x^2 - 9x + 3 = 0$?
I've used a grapher to find approximate solutions at the three places where it crosses the x-axis, and tried my best to work through Cardano's solution ( $t^3 -\frac{31}{3}... | {
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$x^3 + 2x^2 - 9x + 3 = 0$?
I've used a grapher to find approximate solutions at the three places where it crosses the x-axis, and tried my best to work through Cardano's solution ( $t^3 -\frac{31}{3}t + \frac{259}{27} = 0$) but I just can't finish it.
A big part of my problem here is that using the rational root theo... | {
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5. Originally Posted by CaptainBlack
For:
$x^3+a_1x^2+a_2 x+a_3=0$
(snipped)
In which case Tartaglia-Cardano formula reduces to:
$\theta=\arccos(-R/\sqrt{-Q^3})$
$x_1=2\sqrt{-Q}\cos(\theta/3)$
$x_1=2\sqrt{-Q}\cos(\theta/3+2\pi/3)$
$x_3=2\sqrt{-Q}\cos(\theta/3+4\pi/3)$
RonL
Okay, I'll admit: I was not expecting ... | {
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# Do matrices with central symmetry form a group?
Consider the set of $N\times N$ matrices that satisfy the property $$\mathcal{H} = \{H\,|\, H_{ij}=H_{N+1-i,N+1-j}, \det H \neq 0\}$$ or in matrix forms $$\begin{pmatrix}a_{1} & a_{2} & \cdots & a_{N-1} & a_{N}\\ b_{1} & b_{2} & \cdots & b_{N-1} & b_{N}\\ \vdots & \vdo... | {
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Just a quick version of centrosymmetric matrices (thanks to Michael Banaszek for pointing this out):
The matrices you describe in your original question form what is called the centralizer of the involution $$J=\begin{bmatrix} . & . & \dots & . & 1 \\ . & . & \dots & 1 & . \\ & & & & \\ . & 1 & \dots & . & . \\ 1 & . ... | {
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Let $B_{ij}$ be the matrix obtained from $A$ by removing the $i$th row and the $j$th column. That is, we are trying to compare $\det(B_{ij})$ with $\det(B_{N+1-i,N+1-j})$.
I claim that the $(r,s)$ entry of $B_{ij}$ equals the $(N-r,N-s)$ entry of $B_{N+1-i,N+1-j}$.
What is the $(r,s)$ entry of $B_{ij}$?
• If $r\lt i... | {
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So the question comes down to whether the determinant of an $N\times N$ matrix is invariant under the transformation that maps the $(i,j)$ entry to the $(N+1-i,N+1-j)$th entry. This is achieved through a series of row and column exchanges: exchange first row with last row; second row with penultimate row; third row wit... | {
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# Number of elements in cartesian power with a maximum constraint
Problem: I would like to know the number of elements in the cartesian power $X^n$ (cartesian product of one set $X$ by itself, $n$ times) with a maximum constraint: how many elements in $X^n$ have less than $k$ same elements $x$ of the set ($\forall x\i... | {
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Problem: some words are double counted
Strategy 1 for the simple example:
All cases: $|X|^n=3^3=27$ possibilities.
• Let's fix one element $A\in X$ and iterate over $k$.
• Let's fix $k=3$. How many sequences with 3 $A$'s? $\binom{3}{3}=1$ possibility.
• Let's fix $k=2$. How many sequences with 2 $A$'s? $\binom{3}{... | {
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$|X|^n - |X|\sum_{j=k+\lfloor\frac{n-k}{2}\rfloor}^{n} (|X|-1)^{n-j}\cdot \binom{n}{j}$
Same answer than in the question (strategy 1) but $j$ starts not at $k$ but in the middle of the interval $\lfloor k+\frac{n-k}{2}\rfloor$. Justification: by removing all words with $n$, $n-1$, etc. till the middle, we already remov... | {
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# How to normalize data to 0-1 range?
I am lost in normalizing, could anyone guide me please.
I have a minimum and maximum values, say -23.89 and 7.54990767, respectively.
If I get a value of 5.6878 how can I scale this value on a scale of 0 to 1.
• is this the way =(value-min)/(max-min) – Angelo Sep 23 '13 at 15:3... | {
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• I only wonder how the two quite different-looking histograms do illustrate the point of your (correct) answer? – ttnphns Sep 23 '13 at 16:21
• @ttnphns They look only different due to the binning of the histograms. My point however was to show that the original values lived between -100 to 100 and now after normaliza... | {
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• This is correct, but not efficient. It is a linear transformation, so you would precalculate a and b constants, and then just apply newvalue = a * value + b. a = (max'-min')/(max-min) and b = max - a * max – Mark Lakata Sep 23 '13 at 19:18
• Do you know how to cite this? I mean, is there an "original" reference somew... | {
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• There is an important difference between this answer and the already accepted answer. That explained the main idea clearly and directly and then secondarily showed how to do it in one commonly used program. Conversely, you post here only code. While I am happy to believe that this is good code (I don't write PHP) on ... | {
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// JavaScript
function normalize(list) {
var minMax = list.reduce((acc, value) => {
if (value < acc.min) {
acc.min = value;
}
if (value > acc.max) {
acc.max = value;
}
return acc;
}, {min: Number.POSITIVE_INFINITY, max: Number.NEGATIVE_INFINITY});
return list.map(value => {
// Verify that you're not about to divide ... | {
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the answer is right but i have a suggestion , what if your training data face some number out of range ? you could use squashing technique. it will be guaranteed never to go out of range. rather than this
i recommend use this
with squashing like this in min and max of range
and the size of the expected out-of-range ... | {
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# Problem
Let $$f(x)$$ satisfy that $$f(1)=1$$ and $$f'(x)=\dfrac{1}{x^2+f^2(x)}$$. Prove that $$\lim\limits_{x \to +\infty}f(x)$$ exists and is less than $$1+\dfrac{\pi}{4}.$$
# Proof
Since $$f'(x)=\dfrac{1}{x^2+f'(x)}>0$$, $$f(x)$$ is strictly increasing. Thus, $$f(x)>f(1)=1$$ holds for all $$x>1$$, and $$\lim\lim... | {
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The function $$g(x)=\int_1^x\frac{1}{t^2+1}{\rm d}t-\int_1^x \frac{1}{t^2+f^2(t)}{\rm d}t$$ Is strictly increasing and $$g(1)=0 for $$x>2$$ hence $$\lim_{x\to\infty}g(x)\geq g(2)>0$$ so$$\lim_{x\to\infty}g(x)=\lim_{x\to\infty}(\frac\pi4-(f(x)-1))=\lim_{x\to\infty}(\frac\pi4+1-f(x))>0$$
So $$\lim_{x\to\infty}f(x)<\frac\... | {
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# Math Help - easy way of finding line of intersection of 2 planes
1. ## easy way of finding line of intersection of 2 planes
I may be confused so I am hoping someone could verify what I am saying is correct.
To find the line of intersection of 2 planes subtract one from the other so that one of the variables cancel... | {
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. . and we have: . $\begin{Bmatrix}x &=& \text{-}5t - 30 \\ y &=& 3t + 20 \\ z &=& t \end{Bmatrix}$
The reason I'm doubtful is because when I google searched
"line of intersection of two planes" I found a more difficult approach
where the cross product is used to find v. I guess this way is used
to find the vector equ... | {
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Obtain the formula for the following sequence
I can't seem to figure out how to find an algebraic formula for the following sequence of numbers. $$0,\ 1,\ 1,\ 0,\ -1,\ -1,\ 0,\ 1,\ 1,\ 0,\ -1,\ -1,\ 0,\ 1,\ 1,\ 0,\ -1,\ -1,\ 0$$
Can somebody help?
-
Well, I can think of two ways you may have approached this problem... | {
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Addendum: Since you know that it satisfies the recurrence, each term depends only on the previous two terms. Now suppose you know that $f(m) = f(n)$ and $f(m+1) = f(n+1)$ for some $m$ and $n$. Then you know that $f(m+2) = f(n+2)$ also, and, by induction, $f(m+k) = f(n+k)$ for all positive $k$.
But from your enumeratio... | {
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If you accept trigonometry then (as proposed earlier by others)
$$\dfrac{\sin\left(\dfrac {n\pi}3\right)}{\sin\left(\dfrac {\pi}3\right)}\quad \ \text{else}\quad \ \left|\left(n-\dfrac 32\right) \bmod 6-3\right|-\dfrac 32$$
- | {
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# Typical Absolute value inequality
$$\text{How to solve}\quad \frac{\left\vert\,{x + 3}\,\right\vert + x}{x+2} > 1\quad{\large ?}.$$
I tried and wrote two cases, once opening the mod as it is and then the other case opening the mod with a negative sign.
I got the two cases as : $x\in (-\infty,-2)\cup (-1,\infty)$ a... | {
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We need to solve $$\frac{|x+3|+x}{x+2}-1>0$$ or $$\frac{|x+3|-2}{x+2}>0.$$
Now, $x+2=0$ for $x=-2$ and $|x+3|=2$ for $x=-1$ or $x=-5$, which by the intervals method gives the answer: $$(-5,-2)\cup(-1,+\infty).$$
• Wow that is great! Can you explain me what exactly is an interval method and how did you do it? – Tanuj ... | {
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# Orthogonality in non-inner product spaces
I have come across a notion of orthogonality of two vectors in a normed space not necessarily inner product space. Two vectors $x$ and $y$ in a normed space are said to be orthogonal (represented $x\perp y$) if $||x||\leq ||x+\alpha y||,$ for every $\alpha,$ a scalar.
1) Wh... | {
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$\qquad (1,1) \perp_1 (2,0)$ but not $(1,1) \perp_2 (2,0).$
-
Follow up questions: (i) Are these two notions of orthogonality symmetric? $\perp_2$ certainly is, but what about $\perp_1$? (ii) Are these notions invariant under scaling? For example, if $x \perp_1 y$, then $x \perp_1 \beta y$ for any $\beta \in K$. Most ... | {
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The definition you gave is called Birkhoff-James orthogonality and the intuition is the following: suppose you have $x,y\in\mathbb R^2$ and construct a triangle with sides $x$ and $y$. Now let $x$ be fixed and consider the same triangle with $-\alpha y$ instead of $y$. Observe that $||x+\alpha y||$ is the length of the... | {
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-
Concerning your follow-up question (iii) there is the following very nice result: For Birkhoff-James orthogonality it is easy to find examples where $y\perp x$ but $\left\|x\right\|/\left\|x+\alpha y\right\| > 1$ for some real $\alpha$, and so natural to investigate the largest such value $\left\|x\right\|/\left\|x+... | {
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# evaluation of definite integral
#### pantboio
##### Member
I'm struggling for a long time to solve this integral
$$\int_0^\infty e^{-x^2}cos(kx)dx$$
with $k>0$
I know there are a number of ways, but I'm interested in using complex integration. In particular, I believe that we can solve by integrating $e^{-z^2}$ ov... | {
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Kind regards
$\chi$ $\sigma$
Last edited:
#### pantboio
##### Member
my solution:
let $\gamma_R$ be the boundary of the rectangle $[-R,R]\times[0,h]$,for $h$ to determine. Let $f(z)=e^{-z^2}$. Thus
$$\oint_{\gamma_R}f(z)dz=0$$
...but we also have
$$\oint_{\gamma_R}f(z)dz=\oint_{\gamma_1}f +\oint_{\gamma2}f+\oint_{... | {
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$$\frac{1}{2}\sum^{\infty}_{n=0} \frac{(-1)^n k^{2n}\Gamma{(n+\frac{1}{2})}}{(2n)!}$$
$$\frac{1}{2}\sum^{\infty}_{n=0} \frac{(-1)^n k^{2n}\frac{2^{1-2n}\sqrt{\pi}\Gamma{(2n)}}{\Gamma{(n)}}}{(2n)!}\,$$
$$\frac{1}{2}\sum^{\infty}_{n=0} \frac{2\sqrt{\pi}(-k^2)^{n}\Gamma{(2n)}}{4^n(2n)!\Gamma{(n)}}\, \,$$
$$\frac{1}{2}\... | {
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-Dan
#### ZaidAlyafey
##### Well-known member
MHB Math Helper
Let's take a contour such that we include the whole real axis (from -infinity to infinity) and close it off with a semi-circle going from infinity to -infinty. Of course we really have a half-circle with radius R and we take the limit of R as it goes to in... | {
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Here, we can use the general formula
$$\int_{-\infty}^\infty e^{-x^2+bx+c}dx = \sqrt{\pi} e^{b^2/4+c}$$
with $b=ik$ and $c=0$.
$$I =\frac{1}{2} \text{Re} \left[\sqrt{\pi} e^{-k^2/4}\right] = \frac{\sqrt{\pi}}{2} e^{-k^2/4}$$
The formula, I have used can be proved in the following manner:
\begin{aligned} \int_{-\in... | {
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Now if we assume that b is complex the substitution you made is a bit tricky .why ?
Assume that $b= ic$ for simplicity Re(b)=0 .
So let us make the substitution $t= x-\frac{b}{2}= x-\frac{ic}{2}$ but we know that when making a substitution this applies to the bounds of integration as well , but wait how do we do that ?... | {
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# Math Help - Very Stuck. Help on Squares of functions!
1. ## Very Stuck. Help on Squares of functions!
Hi I need some info on how to Sketch the square of functions.
Eg (x+6)(x+3)(x-4) Sketch the square and comment on the number of turning points.
Can someone find a website on this? I have searched google to no ava... | {
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. . The graph is tangent to the x-axis there.
Also the entire graph is above (or on) the x-axis.
Code:
|
* | ** *
** |* *
* * * *| * *
* * * * | * *
- - o*- + - + - o*- + -... | {
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# Find all real p, q, a, and b
#### anemone
##### MHB POTW Director
Staff member
Hi MHB,
Initially, I thought this is another boring high school mathematics problem, but when I started to work on it, I realized I was beaten by it, with equations in variables $a$, $b$, $p$ and $q$ to which I don't see a clear way to ... | {
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so b= (sin t)^(1/10) , q = (cos t)^(1/5)
a = -2 (sin t)^(1/10), p = - ( 1+ (cos t)^(1/5))/2
should be the solution for some t
unless I have missed out something
#### Opalg
##### MHB Oldtimer
Staff member
Find all real $p$, $q$, $a$ and $b$ such that we have $$\displaystyle (2x-1)^{20}-(ax+b)^{20}=(x^2+px+q)^{10}$$... | {
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# DE involving Hyperbolic Trig function (Quick check)
I'm working problem 15 on page 133 of Boyce/DiPrima's Elementary Differential Equations and Boundary Value Problems (10th ed.) Note: this is homework, but it is not graded/turned in.
I have arrived at a solution which I think is correct, but it doesn't match the a... | {
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-
Half-angle formula: dlmf.nist.gov/4.35 – Ron Gordon Jan 29 '13 at 2:18
The answers are equivalent, by the half-angle formula for hyperbolic cosine: $$\cosh^2 \frac{x}{2} = \frac{1 + \cosh x}{2}.$$ This formula is easy to verify from the definition of $\cosh$. So your $C$ is $2c$.
By the way, it's ok for you to drop... | {
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# Probability of Winning a Contest
This is my first question so apologies if its unclear/vague.
There exists a contest with me in it, and $5$ others, thus $6$ people in total, along with $5$ prizes. A person can only win one prize, and once they do, they're out of the contest. Winners are chosen at random.
So, what ... | {
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Using your method, this would give us a total probability of
$$\frac{1}{6}+\frac{5}{6}\left(\frac{1}{5}\right)+\frac{5}{6}\left(\frac{4}{5}\right)\left(\frac{1}{4}\right)+\frac{5}{6}\left(\frac{4}{5}\right)\left(\frac{3}{4}\right)\left(\frac{1}{3}\right)+\frac{5}{6}\left(\frac{4}{5}\right)\left(\frac{3}{4}\right)\left... | {
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-
Calculate it with the probability of the complementary event (not winning a prize), which is: $\frac{5}{6}\cdot \frac{4}{5} \cdot \frac{3}{4}\cdot\frac{2}{3} \cdot\frac{1}{2} = \frac{1}{6}$ , then substract this from 1 to get your probability, which is therefore $1-\frac{1}{6} = \frac{5}{6}$.
-
so - is the answer j... | {
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# I want to calculate the limit of: $\lim_{x \to 0} \left(\frac{2^x+8^x}{2} \right)^\frac{1}{x}$
I want to calculate the limit of: $$\lim_{x \to 0} \left(\frac{2^x+8^x}{2} \right)^\frac{1}{x}$$ or prove that it does not exist. Now I know the result is $4$, but I am having trouble getting to it. Any ideas would be grea... | {
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You might enjoy proving that $$\lim_{x\to\infty}\left(\frac{a^x+b^x}{2}\right)^{1/x}=\max(a,b)$$
Let $f(x) = \left(\frac{2^x+8^x}{2} \right)^\frac{1}{x}$. For $x \to 0$, We have \begin{align*} \log f(x) &= \frac{1}{x} \log \left[ \frac{1}{2}\left(e^{x \log 2} + e^{x \log 8}\right) \right] \\ &= \frac{1}{x}\log\left(1 ... | {
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# Directional Derivative
1. Feb 21, 2016
### RyanTAsher
1. The problem statement, all variables and given/known data
Find the directional derivative of $f$ at $P$ in the direction of $a$.
$f(x,y) = 2x^3y^3 ; P(3,4) ; a = 3i - 4j$
2. Relevant equations
$D_u f(x_0, y_0, z_0) = f_x(x_0, y_0, z_0)u_1 + f_y(x_0, y_0,... | {
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9. Feb 21, 2016
### Ray Vickson
$u_y \neq 4/5$; go back and check your work.
10. Feb 21, 2016
### RyanTAsher
Ah, thank you I missed that. I have remodeled my work, and the solution turns out to be 0, and is correct. Thank you both for your time. | {
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# Open and Closed Sets Discrete Metric
I have 2 contradictory solutions to the follower problem:
The problem:
Let $X$ be an infinite set. For $p \in X$ and $q\in X$ define $$d(x,y)= \begin{cases} 0\qquad&\text{if and only if x=y}\\\ 1&\text{otherwise} \end{cases}$$ Which subsets are closed and which are open?
My "s... | {
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• a closed set does not have to have limit points. you proved every subset is open, and so every subset is closed also. – Forever Mozart Feb 22 '18 at 2:54
• A set is closed if and only if it contains all of its limit points. If the set has no limit points in the first place then this statement is vacuously true. So yo... | {
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## If $$p, s,$$ and $$t$$ are positive integer, is $$|ps - pt| > p(s - t)?$$
##### This topic has expert replies
Legendary Member
Posts: 2276
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### If $$p, s,$$ and $$t$$ are positive integer, is $$|ps - pt| > p(s - t)?$$
by VJesus12 » Wed Aug 18, 2021 7:30 am
00:00
A
B
C
... | {
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CONVERSELY, if x = 4 and y = 6, then we get: |4 - 6| = 4 - 6. In this case |x - y| x - y
Likewise, if x = 5 and y = 20, then we get: |5 - 20| = 5 - 20. In this case |x - y| x - y
And, if x = 0 and y = 1, then we get: |0 - 1| = 0 - 1. In this case |x - y| x - y
Notice the |x - y| = x - y IS true when x > y, and |x - y|... | {
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# Critical Density
1. ### nick1o2
26
I was reading up on critical density, and found the "current" number for it, but can't fine any past records or graphs to show how they have changed over time. Any help?
2. ### dauto
What critical density? If you care enough to actually tell us what you're talking about you migh... | {
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Here's a table of past values of the Hubble time Θ listed in billions of years.
$${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline T (Gy)&\Theta (Gy) \\ \hline 0.473&0.7105\\ \hline 0.566&0.8504\\ \hline 0.678&1.0176\\ \hline 0.811&1.2173\\ \hline 0.971&1.4558\\ \hline 1.162&1.7401\\ \hline 1.390&2.... | {
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8. ### Calimero
256
No, cosmology operates on premise of homogeneity and isotropy, meaning that average density of sufficiently big volume is the same throughout the universe.
9. ### Calimero
256
You can google "wmap" and look at the picture of CMB radiation to see how amazingly universe is uniform. There are tiny f... | {
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Here's the table it prints if you select the range to be from S=11 to S=1, with 20 steps.
That means it will compute and tabulate the universe's history from a time when distances were 1/11 present size up to the present, when distances are their current size i.e. S=1. | {
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$${\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(G... | {
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To find out what the rows mean, click on the link, you will see a sample table, hover the mouse over the blue dots. Then click on "column selection" and you will get more blue info dots telling what the columns mean. And also the "column selection" menu will allow you to select which columns to show. To make that table... | {
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# Prove all other common divisors divide $\gcd$
In the book of Silverman, the below proof is given of the above :
$$a= q_1b + r_1$$ $$b =q_2 r_1+ r_2$$ $$r_1 =q_3r_2 + r_3$$ $$\vdots$$ $$r_{n-3} = q_{n-1}r_{n-2} + r_{n-1}$$ $$r_{n-2} = q_n r_{n-1} + r_n$$ $$r_{n-1} = q_{n+1}r_n + 0$$
But why is $r_n$ the greatest co... | {
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Or may this approach is not proper itself.
• What exactly “appears unconvincing ... from the very start” ? – Martin R Mar 29 '18 at 9:08
• You omitted one step, namely that if you work from the bottom up, you can see that $r_n$ itself divides both $a$ and $b$, so it is a common divisor. After that you then work from t... | {
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So $r_n$ satisfies conditions $(1.)$. Hence $r_n=\gcd(a,b)$
• Basically it is elaboration of the comment of @JaapScherphuis to the OP, and might be (not clear) also incorporates the comment of 'dssknj' (for the 'other direction' part). – jiten Mar 29 '18 at 11:14
• It also makes me feel that there is no 'more' formal ... | {
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