Questions: 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)=frac133 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)=]

$The area (A(r)) (in square meters) of a circular algae colony with radius (r) meters is given by (A(r)=pi r^2). The radius (M(t)) (in meters) after (t) minutes is given by (M(t)=frac133 t). Write a formula for the area (Z(t)) (in square meters) of the algae colony after (t) minutes. It is not necessary to simplify. [Z(t)=]$

Transcript text: 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 $M(t)=\frac{13}{3} t$. Write a formula for the area $Z(t)$ (in square meters) of the algae colony after $t$ minutes. It is not necessary to simplify. \[ Z(t)= \] $\square$

Solution

Solution Steps

Step 1: Define the Radius as a Function of Time

The radius $ M(t) $ of the algae colony is given by the function: \[ M(t) = \frac{13}{3} t \]

Step 2: Substitute the Radius into the Area Formula

The area $ A(r) $ of a circle is given by: \[ A(r) = \pi r^2 \] Substituting $ M(t) $ for $ r $, we get: \[ A(M(t)) = \pi \left(\frac{13}{3} t\right)^2 \]

Step 3: Simplify the Expression

Simplifying the expression for the area as a function of time: \[ A(M(t)) = \pi \left(\frac{169}{9} t^2\right) = \frac{169}{9} \pi t^2 \]