For this problem, we use the formula for radioactive decay which is expressed as follows:
An = Ao (e^-kt)
where An is the amount left after time t, Ao is the initial amount and k is a constant determined experimentally.
First, we calculate for the value of k from the data we have of the half life. We do as follows:
An = Ao (e^-kt)
0.5 = e^-k(5700)
k = 1.21x10^-4 / year
After solving for k, we can use this value to solve for the age of the sample given the percent of the carbon nuclei that is remained after that particular time,
An = Ao (e^-kt)
An/Ao = e^-kt
1 - .10 = e^-1.21x10^-4(t)
t = 878.00 years