Answer
$\color{blue}{\bf{(a) { (A \text{ }\omicron\text{ r} )(t) } = 4\pi{t^2} }}$
$\color{blue}{\bf{(b) }}$$ \color{blue}{\bf{ \text{ the area of the pollution cloud t hours after 8 am }}}$
$\color{blue}{\bf{(c) 64\pi\text{ square miles} }}$
Work Step by Step
The radius $r$ in miles, of a circular cloud of pollution expanding for $t$ hours since $8$am, is modeled by the function $r(t) = 2t$.
The area of the circle is modeled by the formula $A(r)= \pi{r}^2$
$\bf{(a)}$ ${ ( A \text{ }\omicron\text{ r} )(t) } = \pi{(2t)}^2$
$ \color{blue}{\bf{ {(A \text{ }\omicron\text{ r} )(t) } = 4\pi{t^2} }}$
$\bf{(b)}$
$ \color{blue}{\bf{ \text{which is the area of the pollution cloud t hours after 8 am }}}$
$\bf{(c)}$ To find the area of the cloud at noon:
$t$ is the number of hours since $8$ am, and noon is $12$pm
$t=12-8$
$t=4$
${(A \text{ }\omicron\text{ r} )(4) } = 4\pi{4^2} $
$4\pi{(16)} $
$ \color{blue}{\bf{64\pi\text{ square miles}}}$
