a circular oil slick is expanding with radius, r in feet, at time t in hours given by r = 12t - 0.3t², for t…

Explanation:

Step1: Recall area - formula for a circle

The area formula of a circle is $A=\pi r^{2}$, where $r$ is the radius of the circle.

Step2: Substitute the given radius function

We are given $r = 12t-0.3t^{2}$. Substitute this into the area formula: $A=f(t)=\pi(12t - 0.3t^{2})^{2}$. Expand $(12t - 0.3t^{2})^{2}$ using the formula $(a - b)^{2}=a^{2}-2ab + b^{2}$, where $a = 12t$ and $b=0.3t^{2}$. $(12t - 0.3t^{2})^{2}=(12t)^{2}-2\times(12t)\times(0.3t^{2})+(0.3t^{2})^{2}=144t^{2}-7.2t^{3}+0.09t^{4}$. So, $A = f(t)=\pi(144t^{2}-7.2t^{3}+0.09t^{4})$ square - feet.

Answer:

$\pi(144t^{2}-7.2t^{3}+0.09t^{4})\text{ square feet}$