How do I find how much radioactive material is left after time t if I know its half-life?

To answer this question, all you have to know is that the amount of material as a function of time, M(t), is related to the decay constant, λ, by the equation

M(t) = M₀exp(-tλ),

where M₀ is the amount of material you start out with (this could be the mass or the number of particles of the material). λ is expressed in terms of the half-life, t_1/2, as 

λ = ln(2)/t_1/2.

Using these equations and you're known values of M₀ and t_1/2, you can calculate M(t) for any time.

But how do we know these equations are correct? It's all in how the decay rate is defined. We know that the radioactive activity, the amount of material decaying per second, is proportional to the total amount of the material. Hence we can say

dM/dt = -Mλ.

The minus sign is required because we know that the amount of material is being reduced. We can rearrange this equation to the form

dM/M = -λdt

and integrating both sides gives

ln(M) = -λt + c,

where c is a constant. This then gives

M = Aexp(-λt).

Using the fact that M(t=0) = M₀, A must equal M₀, so

M = M₀exp(-λt).