Math Is Fun Forum

An OS written in Java? *shudder*

Nothing can make a faultless operating system, just as no program is ever bug-free. But that doesn't mean you have to use Windows. I use Mac OS X, myself. If you're adventurous, try Linux; it's free, and there are exactly too many choices to choose from.

6. The fraction of the area of the square taken up by the circle is just the ratio of the area of the circle to the area of the square. We know that the areas to be:
πr² = π1² = π
s² = 2² = 4

Answer: π/4., or roughly 3/4.

7. The diagonal of the square will be a diameter of the circle. If the radius is one, the diameter is 2, and we can find the length of the side by trigonometry:

s = 2cos(45) = sqrt(2)

The area is s², or 2. The ratio of their areas is 2/π, or about 2/3. This is the answer.

On the converter, I didn't find any errors, but I do have a suggestion. Would it be possible to make it type from right to left? That's the way binary numbers increase, so it makes more intuitive sense while typing even though on paper it may sound like a bad idea.

For #1, he's almost there. In factoring (3y-1) out, he doesn't need to square it. So, the answer is (y² - 7)(3y - 1)

#2 suffers from the same problem. And, the difference of squares isn't factored quite right. So, the answer is (T - 4)(T + 4)(T - 1).

Me like! Me make noise!

5EC1A1A85E5A81A0
1591121181151210

Shiver me timbers! One it be. Aye!

Page? As in only one? That'd take quite a while to load...

I've wished at times for well-indexed books of formulae. Not books that explain them (I have those), but a quick reference so I can look up stuff I need without having to hunt about for them. I'm sure someone's written one...

Vectors aren't that hard to understand--they're just a major pain to work with.

A vector is a quantity having both magnitude (meaning length) and direction. Think of a vector as an arrow drawn in coordinate space. Vectors don't have position, so you can place them anywhere you like in coordinate space, and as long as you don't altar their direction or magnitude, all is well.

Unit vectors are a convenient way to separate a vector into its component parts. A unit vector has magnitude 1 and is directed along one of the coordinate axes. The unit vector for x is called i, y is called j, and z is called k. They're written with a little hat (^) on top, like this: î (don't know if that'll come through).

So, a vector that has magnitude 5 and is directed 30 degrees above the horizontal has components (4)i and (3)j. If you add them together (using vector addition), they produce the final vector.

You can read more about them on your own. But, you don't need to know much about vectors to solve this problem. You can use the one equation to find the x position, and the other to find the y, and see if, at any one time, the three are the same. Like George, I'm too lazy to work it out for you, but it may involve systems of equations, graphing, and the like.

George,Y's equation is derived from potential energy. Gravitational potential energy is U = mgh. Kinetic energy is K = 1/2 mv². Given that energy is conserved, U = K, so mgh = 1/2 mv². Solving for v gives sqrt (2gh).

This solution assumes that the potential energy due to pressure is the same as gravitational potential energy. It makes a kind of sense, because the fluid pressure in an open-air container arises due to gravity.

This problem can be about as complex as you want it to be. For example, as the water flows out of the tank, the fluid level falls, as does the pressure, so you have to take those rates into account. You can determine the flow rate based on the diameter of the hole and the pressure, but the flow rate also varies with depth.

The arc of the water is influenced by air resistance in a complex way.

Etc., etc., etc. You get the idea.

BTW, one useful little formula is P/ρ, or pressure/density, which gives the energy of a fluid under pressure per unit mass (J/kg).

Sorry if this is incoherent, my brain is fried from the physics test I just took.

and unoftrnatuely, casues us to msis spleilng erorrs all the time! Cool and anyniong at the saem tiem.

I will agree that CSS is broken as a layout tool. But so are tables, as the markup now stands.

I've found from my own experience that CSS layouts are harder to get right, but much, much easier to modify and maintain.

And, from an accessibility standpoint, tables can force you to code elements in a different order than they will appear on the rendered page. So, if somebody's using a screen reader to access your site, they'll be thoroughly bamboozled when things are read out of order.