The area $A(r)$ (in square meters) of a circular algae colony with radius $r$ meters is given by $A(r)=\pi r^{2}$. The radius $M(t)$ (in meters) after $t$ minutes is given by $M(t)=\frac{11}{3} t$.
Write a formula for the area $Z(t)$ (in square meters) of the algae colony after $t$ minutes. It is not necessary to simplify.
\[
Z(t)=
\]
Answer
\(\boxed{Z(t) = 13.4444444444444\pi t^{2}}\)
Step 1 ：We are given two functions, $A(r)$ and $M(t)$. $A(r)$ gives the area of the colony in terms of the radius, and $M(t)$ gives the radius of the colony in terms of time.
Step 2 ：We can substitute $M(t)$ into $A(r)$ to get a formula for the area in terms of time. This means that wherever we see $r$ in the formula for $A(r)$, we replace it with the formula for $M(t)$.
Step 3 ：Substituting $M(t)$ into $A(r)$, we get $Z(t) = \pi (\frac{11}{3}t)^{2}$
Step 4 ：Simplifying the expression, we get $Z(t) = 13.4444444444444\pi t^{2}$
Step 5 ：\(\boxed{Z(t) = 13.4444444444444\pi t^{2}}\)
