ThisGuy

Well, uh, welcome back and stuff, I guess 

So what you'd want to do is find the volume of the sphere, since you're finding out how much paint it can hold. The volume can be found using the equation:

\(V = \frac{4}{3} \pi r^3\)

So, we're given the diameter. Since d = 2r, we can make that r = (1/2)d to find the radius, which would give us 0.75.

Plugging that into our volume equation gives:

\(V = \frac{4}{3} \pi (\frac{3}{4})^3\)

We can simplify that down to: 

\(V = \frac{9}{16} \pi\)

So the paintball holds (9/16)pi units^3 of paint.

My friend Google secretly told me that the volume of a sphere can be found using this formula: \(V = \frac{4}{3}\pi r^3\)

So, to find the volume, plug in the radius of the sphere (or half the diameter, if given) and just solve it from there.

Ah, my words failed me! Thanks for pointing that out :p

Looking at Google, the formula for the volume of a cylinder is \(V = \pi r^2 h\)

Plugging stuff in, we have \(V = \pi * (15/2)^2* 8\)

Solving gives us \(V = 450 \pi\)

or V = 450pi units^2

//edit

I should clarify that r = d/2. That's why I used 15/2.

The +- symbol means that the number can be both positive and negative. So, to elaborate on Meoldy's answer, x = -i and  x = +i. You use it whenever you take the square root of a number, as both (-x)^2 and (x)^2 give the same result, without exception.
You'll see this in the quadratic formula: \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

That just means there are two answers, one where the +- sign is positive and one where it's negative.

NaN, which stands for not a number

My trusty calculator tells me the answer is 28,891,268,902

Rather than doing them for you, I'll guide you through problem number three and have you do the rest. Problem one may look tough, but it's rather simple if you know properties of exponents and logs.

Alrighty, problem number three:

\(x + y = -2\)

\(y+4x=x^2\)

So, using substitution, we want to get one of the equations to start as either y= or x=. I'll just pick the first equation and make it start with  y=.

\(x +y = -2 \Rightarrow x + y -x = -2 - x \Rightarrow y = -2-x\)

So, now our system looks like this:

\(y=-2-x\)

\(y+4x=x^2\)

Now let's work on the second equation in our system. Since we know that y = -2 - x, we can just plug that in whenever we see a y in equation two:

\(y+4x = x^2 \Rightarrow (-2-x) + 4x = x^2 \)

Woohoo! All x's! Let's simplify this to get our x-values now.

\(-2-x+4x = x^2 \Rightarrow -2 + 3x = x^2 \Rightarrow 0 = x^2-3x+2\) -->Now we factor

\(0 = x^2 -3x +2 \Rightarrow 0 = (x-2)(x-1) \Rightarrow 0 = x-2, 0 = x-1 \Rightarrow x=2, x=1\)

So, we've got our two x-values! Neat! Now let's go back to equation 1 and find our y-values using those nifty x-values we just found.

\(y = -2 -x\Rightarrow y = -2 - 2, y = -2 - 1 \Rightarrow y= -4, y = -3\)

So, we've got two x-values and two y-values; now we just put them together and we've got two nice ordered pairs as answers!

Pair 1: (2, -4)

Pair 2: (1, -3)

The other 2 follow the same process as this, just some different algebra to find your values.