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We first use \(y^3-x^3=(y−x)(x^2+xy+y^2)\)

\(x^2 + xy + y^2 = y^3 - x^3 \\ x^2 + xy + y^2=(y-x)(x^2+xy+y^2)\\ y=x+1 \)

From the second equation, we get:

\(yz + 1 = y^2 + z\\ (x+1)z+1=(x+1)^2+z\\ xz+z+1=x^2+2x+1+z\\ x^2+2x-xz=0\\ x^2+x=zx-x\\ x(x+1)=x(z-1)\\ x+1=z-1\\ y=z-1 \)

Concluding:

\(z=y+1\\ y=x+1.\\ \therefore z>y>x\)

They are also consecutive integers. 

In this list, there are \(1998-1732+1\) rows, and therefore, 267 ordered triples. 

I hope this helped,

Gavin

2.cos(-θ)=cos(θ) and sin(-θ)=-(sin(θ))so cos(θ)=4/5 this is a 3-4-5 right triangle so sin(θ)=-(sin(-θ))=3/5. So sin(-θ)=-3/5

3. π=180 and tan(x)=tan(180+x). Thus, the answer is 2/5.

1 and 3 sounds good to me :)

Hint: You can try substituting in the coordinates point and try to see which one fits all the cases.

These problems are from another site that I know! We'll just give you hints, not the full answer.

We treat parts as variables in this question.

Since Kate says she receives 5 parts, that means she receives 5x. 

And, since Rebecca receives  3 parts, that means she receives 3x. 

Adam receives 2 parts, which means 2x.

So, 5x+3x+2x=160, 10x=160, x=16

That means each part is worth £16.

What do you mean by I receive ?? % more than Rebecca?

Do you mean Kate or Adam?

Rebecca: 16*3=£48

If it's Kate, then we'll first have to work out the total amount of money Kate has, which is 16*5=£80

If it's Adam, then we'll first have to work out the total amount of money  has, which is 16*2=£32

To find the percentage increase, we'll have to do: 80-48/48*100=66.6% increase- Kate

Help plz

Euclid's proof:

Suppose that \(p_1=2 < p_2 = 3 < ... < p_r\) are all of the primes. Let \(P = p_1\cdotp_2...pr+1\) and let p be a prime dividing P;

then p can not be any of \(p_1, p_2, ..., p_r,\) otherwise p would divide the difference \(P-p_1p_2...p_r=1\), which is impossible.

So this prime p is still another prime, and \(p_1, p_2, ..., pr\) would not be all of the primes.

\(1. \text{If}\ f(x) = x^2\ \text{and}\ g(x)=\ln(x), \text{compute } f'(1) + g'(1)\)

\(\text{We compute}\ f'(x) = 2x\ \text{and}\ g'(x)=\frac1x,\ \text{so plugging in 1 for both gives us}\ \boxed3. \)

\(2. \int_0^4\frac{dx}{\sqrt{|x-2|}}\)

\(\text{Since the function } \sqrt{|x-2|}\text{ discontinues at } x=2,\\ \text{we can split the integral into two parts and compute separately.}\)

\(\int_0^2\frac{dx}{\sqrt{|x-2|}}=\int_0^2\frac{dx}{\sqrt{2-x}}=-2\sqrt{2-x}|_0^2=2\sqrt2\)

\(\int_2^4\frac{dx}{\sqrt{|x-2|}}=\int_2^4\frac{dx}{\sqrt{x-2}}=2\sqrt{x-2}|_4^2=2\sqrt2\)

\(2\sqrt2+2\sqrt2=\boxed{4\sqrt2}\)

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Hey Radiational!

The best ways to study are to go over the parts of the exam that you are least familiar. Cover your weak points and focus your time where you need it the most. Although you don't need to spend all day prepping parts you already know, you still need to review briefly. Also, take breaks. Don't study 4 hours in a row, go outside and shoot some hoops. This will give you time to relax and calm down. Don't overdose on drugs. Too much coffee or cola is bad for you, as the caffeine is overdosing. In addition, eat healthy with plenty of vitamins and supplements. The day before the test, sleep well and don't study. On the test day, be confident and don't stress out. Good luck 

Volume of clay block =  lxwxh  =6x2x1= 12 in^3

Cylinder volume = pi r^2 h  =    pi (2)^2 7 = 87.965 in^3

87.965/12= 7.33 clay blocks ~ 8 clay blocks

https://web2.0calc.com/questions/help-please_16499

Nice job on overcoming this problem! Or in this case remembering how to do it.

If the question glitches report the problem to a moderator, I reccomend one that's online.

The most I can do is link you to a guide if that's okay. 

https://www.khanacademy.org/math/algebra/introduction-to-algebra/alg1-dependent-independent/e/dependent-and-independent-variables 

Complete this and get back to me to tell me if it helped or not.

@guest Can you walk us through on how you got that answer?

Most fans of team sports are intuitively aware of the “home court” or “home field” advantage and the disadvantage for away games.  This implies the winning or losing is dependent on the home or away condition.   

Here is a mathematical test for dependent/independent using the chart data.   

\(\text {If this is independent then } \\ P(L|A)=P(L) \tiny \text{ Read as “The probability of (L)osing, given the game is (A)way = the probability of (L)osing. }\\ P(0.3|0.4) \stackrel{?}= P(0.65)\\ 0.75 \neq 0.65 \tiny \text{ These are not equal, so the events are dependent. }\\\)

A notable increase in the probability for losing the away games. 

GA

Method 1: Logic

One the sphere, there exists a point A and its antipodal point, A'. This means that A' is the point with the farthest distance from A, the 3D version of the diameter, which means when you connect A and A', the line contains the center of the sphere. After one chooses the three points, W, X, Y, the region to place Z is the spherical traingle W'X'Y' opposite WXY. It is only then, the tetrahedron would contain the center of the sphere. This means that the probability the tetrahedron would contain the center of the sphere is the expected area of the triangle W'X'Y', which has the same expected area of WXY and 7 other triangles: WXY, W'XY, W'X'Y, W'X'Y', WX'Y, WX'Y', WXY', W'XY'. Since there are 8 such triangle, and the total area is set 1, the expected area is 1/8.  The probabilty is 1/8. 

Method 2: Mathematical Proof

We can proof the logic using the notion of barycentric coordidates: https://nicoguaro.github.io/posts/putnam_prob/ 

I think the website does a great job explaining the process, better than I can at least. If you have any questions though, about the website or my logic, ask away!

I hope this helped,

Gavin

1

The normal height of a sin or cos fxn is from 1 to -1  for a total height of  2      the requested fxn has a height of 10-(-4)=14   which is 7 times higher, so it is a wave multiplied by 7

7 cos    or   7 sin         it is shifted UP by 3  ...so add 3

7cos() +3    or    7sin() + 3

Period of sin or cos wave is 2pi      for a period of  pi/2   you  have to multiply by 4

it is 3 at x=0     so it is a sin wave

7 sin(4x) + 3

2.Tan = sin/cos    if you add pi, both the sin and cos become opposite (negative), so tan still equals   2/5

2/5

3.

The value of sin is between 1 and -1 (inclusive)   so this question is invalid.

  If you MEANT  - sqrt(3) /2    then    4pi/3  or  5pi/3  would work

Need diagram.

This exact question is answered just a few more questions down the list.....

Circumference = pi * d = pi * 2 * r

Fraction of circumference =  18 pi  / (pi*2*r)    (is also fraction of 360 degrees)

Fraction of 360 degree circle   360 *  18 pi /(pi*2*r)

=108 degrees