A corrugated metal pipe $($n $=0\mathrm{.}022)$ was originally designed to carry runoff from a $5-$year storm while flowing half full. However, due to an increase in rainfall over the last $50$ years, it will be replaced with a steel pipe $($n $=0\mathrm{.}013).$ The steel pipe is designed to carry runoff when flowing three$-$quarters full and must handle a flow rate that is $30\%$ higher than the original flow rate of the corrugated metal pipe. Will the steel pipe need to have a larger or smaller diameter than the corrugated metal pipe? Calculate the percentage change in diameter