The surface area $S(r)$ (in square meters) of a spherical balloon with radius $r$ meters is given by $S(r)=4 \pi r^{2}$.
The radius $P(t)$ (in meters) after $t$ seconds is given by $P(t)=\frac{4}{3} t$.
Write a formula for the surface area $N(t)$ (in square meters) of the balloon after $t$ seconds.
It is not necessary to simplify.
Answer
\(\boxed{N(t) = \frac{64}{9} \pi t^{2}}\)
Step 1 ：Given the surface area formula for a sphere with radius r: \(S(r) = 4 \pi r^{2}\)
Step 2 ：Given the radius formula for the balloon after t seconds: \(P(t) = \frac{4}{3}t\)
Step 3 ：Substitute the radius formula into the surface area formula: \(N(t) = 4 \pi (\frac{4}{3}t)^{2}\)
Step 4 ：\(N(t) = 4 \pi (\frac{16}{9}t^{2})\)
Step 5 ：\(N(t) = \frac{64}{9} \pi t^{2}\)
Step 6 ：\(\boxed{N(t) = \frac{64}{9} \pi t^{2}}\)
