Answer
$6$ significant digits.
Work Step by Step
See the table below, where we calculate $P(50)=180(b)^{50}$
for the year 2010, which is 50 years after 1960.
We start from 12 significant digits for b, and then reduce to 11, 10, etc.
We see that up to 6 significant digits, the rounded result is 309, which is what we want.
When we drop to 5 significant digits, the rounded result is $310$.
So, our answer is that we need
$6$ significant digits.
