I think there's a misunderstanding in the question. There seems to be a missing formula for the radius after t seconds symbolized by "Pt". In order to provide a formula for the surface area N(t), we need that information first.
However, assuming we have the formula for the radius after t seconds as Pt, here's how we can proceed:
1. We are given the surface area S(r) of a spherical balloon, which is represented by the formula S(r) = 4 * pi * r^2. This formula calculates the surface area of a sphere using the radius r.
2. Next, we're asked to find the formula for the surface area N(t) of the balloon, but using the time t instead of the radius r. To do so, we'll have to substitute the radius r in the surface area formula with Pt, which represents the radius of the balloon after time t.
So, by replacing r in the original formula with Pt, we get:
N(t) = 4 * pi * (Pt)^2
Without the specific formula for Pt, we can't simplify it further than this. However, this formula allows you to calculate the surface area of the balloon after t seconds, once you know how the radius changes with time.