Certainly! Let's derive the formula for the area [tex]\( Z(t) \)[/tex] of the algae colony after [tex]\( t \)[/tex] minutes step-by-step.
1. Understand the formulas:
- The area of a circle with radius [tex]\( r \)[/tex] meters is given by [tex]\( A(r) = \pi r^2 \)[/tex].
- The radius [tex]\( r \)[/tex] of the algae colony after [tex]\( t \)[/tex] minutes is given by [tex]\( M(t) = \frac{10}{3} t \)[/tex].
2. Substitute the expression for the radius [tex]\( r \)[/tex] from [tex]\( M(t) \)[/tex]:
- Since [tex]\( r = M(t) = \frac{10}{3} t \)[/tex], we will substitute [tex]\( \frac{10}{3} t \)[/tex] for [tex]\( r \)[/tex] in the area formula [tex]\( A(r) = \pi r^2 \)[/tex] to find [tex]\( Z(t) \)[/tex], the area as a function of time.
3. Substitute [tex]\( M(t) \)[/tex] into [tex]\( A(r) \)[/tex]:
- Start with the area formula [tex]\( A(r) = \pi r^2 \)[/tex].
- Substitute [tex]\( r = \frac{10}{3} t \)[/tex]:
[tex]\[
Z(t) = \pi \left( \frac{10}{3} t \right)^2
\][/tex]
4. Write the final expression:
- The formula for [tex]\( Z(t) \)[/tex], the area of the algae colony after [tex]\( t \)[/tex] minutes, is:
[tex]\[
Z(t) = \pi \left( \frac{10}{3} t \right)^2
\][/tex]
So, the area [tex]\( Z(t) \)[/tex] of the algae colony after [tex]\( t \)[/tex] minutes is given by:
[tex]\[
Z(t) = \pi \left( \frac{10}{3} t \right)^2
\][/tex]
This is the required formula for the area of the algae colony as a function of time.
