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Another interesting answer.  Thank you Chris.  

Often eigenvectors are normalised to unity.  If you do that here remember, when calculating the normalisation constant, to multiply the complex number component by its complex conjugate; don't just square it!

Thanks Rom,

That gives me several things to think about.  I had no idea how to do that one.

Plus, I have never heard the word 'apothem' before.

I have never seen that area formula before either but I am sure I could derive it easily enough.

How interesting!  

Each person will get a half, that is $$\frac{1}{2}$$ but 'every' won't get any because she wasn't one of the people getting some donut.

you need to be much more specific.

2n-(3+4)n+6n+1=n+1

Let the number be N then your expression will be:

6N-2

Thank you Rom

14 = 1400%

Look at a regular polygon of n sides.

The area is given by

$$A_n=\dfrac s 2 a n$$

s is the length of a side

a is the length of the apothem

n is the number of sides

I leave it to you to work out that for a given radius r

$$\dfrac s 2 = r \sin\left(\dfrac \pi n\right)$$

$$a=r \cos\left(\dfrac \pi n\right)$$

so

$$A_n=r^2 n\sin\left(\dfrac \pi n\right)\cos\left(\dfrac \pi n\right)$$

You can show that

$$\displaystyle{\lim_{n\to\infty}}n \sin\left(\frac \pi n\right)\cos\left(\frac \pi n\right)=\pi$$

so

$$\displaystyle{\lim_{n\to \infty}}A_n=\pi r^2$$

This limit corresponds to a regular polygon of increasingly many but smaller sides which taken to infinity represents a circle and thus you get

$$A_{circle}=\pi r^2$$

Why is it not possible to solve log(x, -3)= ?

I'm not familiar with this notation, but we can't take a log of a negative number, nor can we use a negative number as a log base - if this is what you're asking???

To see the first one, suppose I had log (-23). In exponential terms, I want to know what exponent I can append to 10 to give me -23. Answer......there is none. 10 raised to any power is a positive number.

To see the second one, suppose we used (-10) as a base. Now, suppose I wanted to find  log_-10 10.

This says, in exponential form, what power do I put on (-10) to raise it to 10?? Answer..... there is none.

Note (-10)^0 = 1 but  (-10)^1 = -10 and (-10)^2 = 100. Note how the answers "oscillate" back and forth. In other words, (-10) raised to 0 is actually greater than (-10) raised to the next integer power !!!! Notice also, that (-10)^3 would, again, be negative (-1000) - and less than (-10)^2 !!!!

Thus, we can't use a negative number as a log base, and we can't take the log of a negative number (or "0,"

either !!!).........I hope this answers your question.......

this question has been answered

$$P V = n R T$$

$$V1 = \dfrac {n R T}{P}$$

$$V2 = \dfrac {n R (2 T)}{(2P)}=\dfrac{n R T}{P}=V1$$

$$V2=V1$$

Do you mean on the site calculator?

3 and a half divided by 6 and 2 thirds

(3+1/2) $$\div$$ (6+2/3)=

The surface area of the "curved" side of the cylinder is given by 2*(pi)*r*h

By "missing a circle," I assume you mean it only has one closed end.

If the cylinder has only one closed end, we must add the area of this = (pi)*(r^2)

And that's it......

how to divide a problem with a fraction and whole number

Well......let's do a couple of examples

Suppose we have 2/3  divided by 5

First, write the "5" as a fraction = 5/1

Then, the rule is " Keep"  "Change" "Flip'

"Keep" the first fraction   2/3

"Change"  Division to Multiplication

"Flip" the last fraction.......putting it all together, we have

2/3 X 1/5 = 2/15    (Note that we might have to reduce the final answer....Some people do this BEFORE they multiply)

Now.....suppose we have 5 divided by 2/3

So, applying the steps above , we have

5/1 x 3/2 = 15/2

Here's some websites where you can practice :

I thought i just wrote a long answer for this?

'I did, I did see a tweety bird.'  Don't mind me, that is just a line out of an ancient cartoon. 

http://web2.0calc.com/questions/what-is-the-area-of-a-regular-square-with-the-radius-as-3-round-to-one-decimal-place-nbsp-also-how-do-we-report-spammers-nbsp

The surface area of the "curved" side of the cylinder is given by 2*(pi)*r*h

If the cylinder also has a top and bottom, we must add the area of these two things = 2*(pi)*(r^2)

See if you can take it from here.........

Hi SpawnofAngels,

Squares do not have a radius so your question does not make much sense.

Only circles have a radius.

Do you mean that the square is circumscribed by the circle OR that the circle is circumscribed by the square OR something else. If you do not know what circumscribed means, look here.

how do you identify the graph of the function y=-(x-3)^2+2 ?

This is a parabola that opens "downward" with its vertex at the point (3,2).

$$250^{1/42}\approx1.140496114$$

250^(1/42)  = 1.1404961136900191

If you raised the answer to the 42nd power, it would return "250."   (theoretically)

What is the solution to (y(2-y))/(1-y)^2=1.7777e^(0.26(10))

So we have.... (2y  - y^2) / (1 -y)^2 = 1.7777e^(2.6)

This one would require some work to solve, so I'll take adantage of the calculator on this site to do the "heavy lifting."

We get two solutions (rounded)

y ≈ .799   and  y ≈ 1.2003

A parallelogram is a 4-sided shape formed by two pairs of lines. Opposite sides are equal in length and opposite angles are equal in measure. To find the area of a parallelogram, multiply the base by the height. The formula is:
where is the base, is the height, and · means multiply.
The base and height of a parallelogram must be . However, the lateral sides of a parallelogram are not perpendicular to the base. Thus, a dotted line is drawn to represent the height.

I believe this is what you needed.....

how can I prove that pi times radius squared is the area of a circle

Mmmmmm....this is an interesting one.

Let's start by inscribing a polygon of some "n" sides in a circle. It can be shown that, as the number of sides increase (to some "large" value), the perimeter of this polygon is almost the same as the circle's circumference ≈ 2*pi*r. And since we have n sides, the length of each side is just ≈ (2*pi*r) / (n)

Now, if from the center of the circle, we draw a radial line (r) to each vertex of the polygon, we will form "n" triangles. The base of each triangle is just the side length of the polygon (given in the previous paragraph). And we can also show that, as the number of sides of the polygon increase to some "large number," the height of each triangle aproaches "r" (the radius of the circle).

Therefore, the area of each triangle is (1/2)*(b)*(h) = (1/2)*[(2*pi*r) / (n)]*(r) =

pi*r^2 / (n).

And since we have n triangles, then the total area of all the triangles = (the number we have) * (the area of each one) =

(n)*(pi*r^2) / (n) =

(pi)*r^2........