Solve the Problem $S$ (In Square Inches) of a Cylindrical Pipe with Length 12

Essay

Solve the problem.
-The surface area $S$ (in square inches) of a cylindrical pipe with length 12 inches is given by $S ( r ) = 2 \pi r ^ { 2 } + 24 \pi r$ , where $r$ is the radius of the piston (in inches). If the radius is increasing with time $t$ (in minutes) according to the formula $r ( t ) = \frac { 1 } { 6 } t ^ { 2 } , t \geq 0$ , find the surface area $S$ of the pipe as a function of the time $t$ .