Math Problems

If a and b are the sides of the rectangle and r is the radius of the circle, then:
r = sqrt(a^2+b^2)/2

Then the area of the circle is:
pi*r^2 = pi*(a^2+b^2)/4

Replacing b with 200/a:
Area = (pi/4)(a^2+(200/a)^2) = (pi/4)(a^2+40000/a^2)

Differentiating area wrt a:
(pi/4)(2a-80000/a^3)

Setting this equal to zero, we can get rid of the pi/4 and are left with:
2a-80000/a^3 = 0

Getting a common denominator & solving for a:
(2a^4-80000)/a^3 = 0
2a^4-80000 = 0
a = sqrt(200)

Which makes sense because the area of the circle would be minimized when radius is minimized. And radius is minimized when a = b (i.e. a square).

Edit: Whoops. Somehow I didn't see the post above mine. O.o He's right too!

EDIT2: Anyone wanna help me out with a math problem I got stuck on?

Find all integers n with the property that



is also an integer.

I spent like 2 hours trying to algebraically manipulate it and practically got nowhere. I then used my calculator to find out that n=49 is a solution. I haven't tried finding any other solutions, as "I used my calculator" is not a mathematical proof at all.