How close does an alpha particle get to a gold nucleus?

Use conservation of energy.  The speed is low enough that you can get away with classical (non-relativistic) mechanics.  We can neglect the impact of the gold's electrons if we end up near the center of their orbitals.

Initial kinetic energy = 1/2 mv^2
=
Final electrostatic potential energy = k q1 q2 / r

Solve for the distance:
r = 2 k q1 q2 / mv^2

k is coulomb's constant--look it up
q1 and q2 are the charges of the alpha and the gold nucleus, which are given in terms of the fundamental charge, e--look that up
m is the alpha particle's mass, which is given
v is the speed at which it is fired, also given

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