Alan

If rotations are not allowed, but reflections are, then I think Melody's formula is correct.  See the actual arrangements below for two couples and for three couples:

Not that I want to overcomplicate things here, but, if a wife sits to the left of her husband does that count as a different arrangement from sitting to the right of him or not?

Ok.  Didn't read Melody's reply carefully enough!  Thanks for pointing this out Anonymous.

If you want it integrated to find y as a function of x then:

We want the times for which (√3)sin(2t) + cos(2t) = 0.  We can rearrange this to get tan(2t) = -1/√3  (by dividing through by √3*cos(2t) and moving one term to the other side of the = sign).

Now tan(pi/6) is 1/√3, but it is negative in the second and fourth quadrants, so we must have tan(n*pi - pi/6) = -1/√3, where n is a positive integer and the angles are in radians (in degrees pi is 180 and pi/6 is 30).  

This means 2t = n*pi - pi/6 or t = n*pi/2 - pi/12.

With the calculator on the Home page here you can use the plot command.

I guess 32) might be:

$$3\sqrt[3]{5y^3}\times 2\sqrt[3]{50y^4}$$

Multiply the terms outside the surds together, and multiply the terms inside the surds together:

$$6\times \sqrt[3]{250y^7}$$ 

Now you want to express the terms inside the surd as something-cubed multiplied by whatever is left over and can't be expressed as a nice cube!

$$250=2\times 5^3\\
y^7=y\times (y^2)^3$$ 

So:

$$6\sqrt[3]{250y^7}=6\sqrt[3]{2\times 5^3\times y\times (y^2)^3}=6\times5\times y^2\times \sqrt[3]{2y}$$

or just:

$$30y^2\sqrt[3]{2y}$$

But you must use brackets more to ensure your meaning is clear and unambiguous Nataszaa!

This is an empirical formula for the wind-chill temperature (in °F) where the ambient temperature is T°F and the wind-speed is V miles/hr.  It's normally written as:

$$$$WCT=35.74+0.6215\times T-35.75\times V^{0.16}+0.4275\times T\times V^{0.16}$$

(Notice that 0.16 = 4/25)

ln(1) is 0  (in fact log_b(1) is zero, whatever the base, b, is because b⁰ = 1; and when log_b(x)=y, this means, by definition, that b^y = x.  In your case b = e, x = 1 and y = 0).  You should be able to do the rest.

The area is just length* width = 2√2 square units.  Because √2 is irrational (i.e. cannot be represented exactly by the ratio of two whole numbers) then so is the area.