A centrifugal pump with a $30-$inch diameter impeller and a rotation speed Its $1,100$ revolutions per minute impeller pumps a water flow of $2,605$ gallons per second. A second pump, geometrically similar to the first and manufactured by the same manufacturer, also operates with an impeller rotation speed of $1,100$ revolutions per minute, but pumps a water flow rate of $6\mathrm{.}175$ gallons per second. Determine the diameter $($inches$)$ of the impeller of the second pump.