For radioactive decay we use the formula:

A(t) = A0*e^(-rt)

where A(t) is the amount left after time t, A0 is the intitial amount, r is the rate of decrease, t is the time

r = ln2/t, where t is the half-life1) Radioactive decay problem: What percent of a present amount of radioactive radium (226Ra) will remain after 900 years? (Half-life: 1599 years)

=> r = ln2/1599 = 0.000433487

we want A(t) when t is 900, that is, we want A(900)

A(900) = A0 *e^(-0.000433487(900)) = A0(0.676963)

so A(900) = (0.676963)A0

so A(900) is 67.69% of A0. this is the percent remaining

assume we start with 1 unit of the radioactive substance, then after 1 year, 0.9957 units are left.2) Radioactive decay problem: Find the half-life of a radioactive material if after 1 year 99.57% of the initial amount remains.

using A(t) = A0*e^(-rt)

we have A(1) = 0.9957 = 1*e(-r(1))

=> e^-r = 0.9957

=> ln(e^-r) = ln(0.9957)

=> -r = ln(0.9957)

=> r = - ln(0.9957)

=> r = 0.004309271

now t = ln2/r, where t is the half life

=> t = ln2/0.004309271

=> t = 160.8502 years