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at first you would think 4, but if we do tan-1 (sin(40) - cos{36)) we get - 9.43, now we take that and add both 40 and 36 from to it :

(-9.43+40)-(-9.43+36) we get 4, and as to tackle the problem from its roots we root it giving us 2, jojo has 2 banans, 3 if she has a boyfriend/husband

Hi Tetre,

The word trapezoid means different things in different countries, could you plase give a definition that we should use?

I'll just tell what I think you mean.

AB and DC are parallel.  

This means that AEB and DEC are similar triangles 

so

the ratio of the height of ABE and DEC will be   8:17

so  Area of  AEB = 0.5*8*8k       and area of DEC = 0.5*17* 17k    

and area of  ABCD = 0.5*17* 17k  -   0.5*8*8k

ratio of the area of triangle EAB  to the area of trapezoid ABCD

\(=\frac{0.5*8*8k}{0.5*17*17k\;-\;0.5*8*8k}\\ =\frac{8*8k}{17*17k\;-\;8*8k}\\ =\frac{8*8}{289\;-\;64}\\ =\frac{64}{225}\\\)

{4 x = 6 - 2 y | (equation 1)
x = y + 7 | (equation 2)
Express the system in standard form:
{4 x + 2 y = 6 | (equation 1)
x - y = 7 | (equation 2)
Subtract 1/4 × (equation 1) from equation 2:
{4 x + 2 y = 6 | (equation 1)
0 x - (3 y)/2 = 11/2 | (equation 2)
Divide equation 1 by 2:
{2 x + y = 3 | (equation 1)
0 x - (3 y)/2 = 11/2 | (equation 2)
Multiply equation 2 by 2:
{2 x + y = 3 | (equation 1)
0 x - 3 y = 11 | (equation 2)
Divide equation 2 by -3:
{2 x + y = 3 | (equation 1)
0 x+y = (-11)/3 | (equation 2)
Subtract equation 2 from equation 1:
{2 x+0 y = 20/3 | (equation 1)
0 x+y = -11/3 | (equation 2)
Divide equation 1 by 2:
{x+0 y = 10/3 | (equation 1)
0 x+y = -11/3 | (equation 2)
Collect results:
Answer: | x = 10/3        and        y = -11/3

x=7+y, 4x=6-2y = {x=(10/3), y=-((11/3))}

Yes, there is a loop :)

\(\int e^{5x}sin(3x)dx=[\frac{e^{5x}}{5}\cdot sin(3x)]-\int \frac{e^{5x}}{5}\cdot3cos(3x)dx\\ \int e^{5x}sin(3x)dx=\frac{e^{5x} sin(3x)}{5}-\frac{3}{5}\int e^{5x}cos(3x)dx\\ \int e^{5x}sin(3x)dx=\frac{e^{5x} sin(3x)}{5}-\frac{3}{5}\left[(\frac{e^{5x}cos(3x)}{5})-\int \frac{-3}{5} e^{5x}sin(3x)dx\right]\\ \int e^{5x}sin(3x)dx=\frac{e^{5x} sin(3x)}{5}-\frac{3e^{5x}cos(3x)}{25}- \frac{9}{25}\int e^{5x}sin(3x)dx+c\\ \int e^{5x}sin(3x)dx+ \frac{9}{25}\int e^{5x}sin(3x)dx=\frac{e^{5x} sin(3x)}{5}-\frac{3e^{5x}cos(3x)}{25}+c\\ \frac{34}{25}\int e^{5x}sin(3x)dx=\frac{5e^{5x} sin(3x)}{25}-\frac{3e^{5x}cos(3x)}{25}+c\\ \int e^{5x}sin(3x)dx=\frac{5e^{5x} sin(3x)}{34}-\frac{3e^{5x}cos(3x)}{34}+c\\ \int e^{5x}sin(3x)dx=\frac{e^{5x} (5sin(3x)-3cos(3x))}{34}+c \)

Take the integral:
 integral e^(5 x) sin(3 x) dx

For the integrand e^(5 x) sin(3 x), integrate by parts, integral f dg = f g - integral g df, where 
 f = sin(3 x), dg = e^(5 x) dx, df = 3 cos(3 x) dx, g = e^(5 x)/5:

 integral e^(5 x) sin(3 x) dx = 1/5 e^(5 x) sin(3 x) - 3/5 integral e^(5 x) cos(3 x) dx
For the integrand e^(5 x) cos(3 x), integrate by parts, integral f dg = f g - integral g df, where 
 f = cos(3 x), dg = e^(5 x) dx, df = -3 sin(3 x) dx, g = e^(5 x)/5:

 integral e^(5 x) sin(3 x) dx = 1/5 e^(5 x) sin(3 x) - 3/25 e^(5 x) cos(3 x) - 9/25 integral e^(5 x) sin(3 x) dx
Add 9/25 integral e^(5 x) sin(3 x) dx to both sides:

34/25 integral e^(5 x) sin(3 x) dx = 1/5 e^(5 x) sin(3 x) - 3/25 e^(5 x) cos(3 x) + constant
Multiply both sides by 25/34:

 integral e^(5 x) sin(3 x) dx = 25/34 (1/5 e^(5 x) sin(3 x) - 3/25 e^(5 x) cos(3 x)) + constant
Which is equal to:
Answer: | = 1/34 e^(5 x) (5 sin(3 x) - 3 cos(3 x)) + constant

300 is 150%

There is an arithmetic of infinite numbers - search for "transfinite numbers".

The "smallest" infinite number is the number of natural numbers (1, 2, 3 etc), and is generally represented by the symbol  \(\aleph_0\)        .

Multiples and finite powers of this are still  \(\aleph_0\)    !

However, the number of Real numbers (including irrational) is a "bigger" infinity!  

Also   \(\aleph_0^{\aleph_0}\)     is bigger!

Transfinite arithmetic is weird and wonderful!

You should read this article. Also, look up the mathematician Georg Cantor who spent a great deal of his life on such matters as infinity.

https://en.wikipedia.org/wiki/Infinity

I agree, but my visualization of infinity in this theory is the amount of possible numbers, or in this case, numbers, decimals, and fractions.

Thanks for your opinion!

Hi :)

I always like to see people thinking about numbers the way you do but infintiy^2 has no real meaning. I mean it is no bigger than infinity.  You can square huge numbers and get an even bigger number but infinity is not a huge number. It is a concept. 

I'm visualizing the concept of infinity as the theoretical "number" of possibilities for a number or decimal, or "numerical infinity".

But I agree with you 

Infinity is NOT a number but a CONCEPT or IDEA, such as space stretching into every direction FOREVER....etc. !!!!.

As follows:

I.  Melody has shown that the result is pi/4.  This is 0.79 to two dp.

II.  Use the fact that \(e^{i\pi/2}=i\)   so \(i^i=(e^{i\pi/2})^i\rightarrow e^{i\times i\pi/2}\rightarrow e^{-\pi/2}\)

Hence \(\chi=5+ie^{-\pi/2}\)

\(Real(\chi)=5.00\) to two dp   and  \(Imag(\chi)=0.21\)to two dp

III.  Separate variables and use the fact that \(\frac{1}{\psi(1-\psi)}=\frac{1}{\psi}+\frac{1}{1-\psi}\)the integrals are then straightforward.  Remember to include a constant, k, say, because the integrals are indefinite.  You then have two unknowns, \(\beta\) and k.  Use the two given conditions to find them.  Then you can determine \(\psi(4)\).  (You should find \(\psi(4) = 0.99\) to two dp).

(I would have supplied more detail, but the image uploading process isn't working at present!)

\(1+1=\sqrt[35]{34359738368}\)

Your  method would work if you did your math right: (5/4)*80 = 100

Hi Geno,

This still does not make sense to me.

Ther must be more teams playing or else the percentages do have to add to 100% and if other teams are playing and there are 180 games in total then no team will actually get to play 180 games will they. 

So like i said, id does not make sense to me :/

2y^2-12y-17=x

This is a sideways parabola (because it is the y that is squared)

it opens out in the positive direction because   the number in front of the y^2 is positive (+2)

When y=0, x=-17

So it must have 2 real roots, if you sketch what I have said so far you will see why.

\(\triangle = b^2-4ac = 144-136=8\)

8 is not a perfect square so there are 2 irrational roots.

The axis of symmetry is   y=-b/2a = 12/4 = 3

So the roots, (y intercepts), are,    

\(x=\frac{--12\pm\sqrt8}{4}\\ x=\frac{12\pm2\sqrt2}{4}\\ x=\frac{6\pm\sqrt2}{2}\\ \text{So it is a sideways parabola that passes through}\\ (\frac{6+\sqrt2}{2},0),\;\;\;(\frac{6-\sqrt2}{2},0),\;\;\;and \;\;\;(-17,0)\)

At the vertex y=3 sub that in to get the value of x

There are 60 seconds in a minute.......30  and above, round up to next minute, else, round down

So......

48°16'