To find the surface area [tex]\( N(t) \)[/tex] of the balloon after [tex]\( t \)[/tex] seconds, we start with the given formulas:
1. The surface area [tex]\( S(r) \)[/tex] of a spherical balloon with radius [tex]\( r \)[/tex] meters is given by:
[tex]\[ S(r) = 4 \pi r^2 \][/tex]
2. The radius [tex]\( P(t) \)[/tex] in meters after [tex]\( t \)[/tex] seconds is given by:
[tex]\[ P(t) = \frac{5}{3} t \][/tex]
We need to substitute the expression for [tex]\( P(t) \)[/tex] into the surface area formula.
First, substitute [tex]\( P(t) \)[/tex] for [tex]\( r \)[/tex] in the surface area formula:
[tex]\[ S(P(t)) = 4 \pi \left( P(t) \right)^2 \][/tex]
Next, replace [tex]\( P(t) \)[/tex] with [tex]\( \frac{5}{3} t \)[/tex]:
[tex]\[ S(P(t)) = 4 \pi \left( \frac{5}{3} t \right)^2 \][/tex]
Now, compute the square of [tex]\( \frac{5}{3} t \)[/tex]:
[tex]\[ \left( \frac{5}{3} t \right)^2 = \left( \frac{5}{3} \right)^2 t^2 = \frac{25}{9} t^2 \][/tex]
Substitute this result back into the surface area formula:
[tex]\[ S(P(t)) = 4 \pi \left( \frac{25}{9} t^2 \right) \][/tex]
Simplify the expression inside the parentheses:
[tex]\[ S(P(t)) = \frac{100}{9} \pi t^2 \][/tex]
Therefore, the formula for the surface area [tex]\( N(t) \)[/tex] after [tex]\( t \)[/tex] seconds is:
[tex]\[ N(t) = \frac{100}{9} \pi t^2 \][/tex]
So, the completed formula is:
[tex]\[ N(t) = \frac{100}{9} \pi t^2 \][/tex]
