math question on a cell

Originally posted by wormer
assuming that a DNA is a perfect cylinder and the nucleus is a perfect sphere. Prove mathematically that that DNA can fit into a nucleus.

Suppose a cylinder of radius a and length 2h just fits inside a sphere of radius r. The cylinder touches the sphere to form two circles of radius a with formulae x^2 + y^2 = a^2, z = +/- h. So we can use Pythagoras' theorem to relate a, h and r. Imagine the triangle formed by the origin, the point h on the z axis and a point on the circle of contact. We have that:

r^2 = h^2 + a^2

So the cylinder will fit in the sphere if:

h < √(r^2 - a^2)

and

a < √(r^2 - h^2)

The length l of the cylinder is 2h so we can write these conditions as

l < 2√(r^2 - a^2)

and

a < √(r^2 - (l^2)/4)