$1.$ Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains $200$ L of a dye solution with a concentration of $\backslash (1\backslash$mathrm$\{$~g$\}/\backslash$mathrm$\{$L$\}\backslash ).$ To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of $\backslash (2\backslash$mathrm$\{$~L$\}/\backslash$mathrm$\{$min$\}\backslash ),$ the well$-$stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches $\backslash (1\backslash \%\backslash )$ of its original value.

$2.$ A tank originally contains $100$ gal of fresh water. Then water containing $\backslash (\backslash$frac$\{1\}\{2\}\backslash$mathrm$\{$lb$\}\backslash )$ of salt per gallon is poured into the tank at a rate of $\backslash (2\backslash$mathrm$\{$gal$\}/\backslash$mathrm$\{$min$\}\backslash ),$ and the mixture is allowed to leave at the same rate. After $10$ min the process is stopped, and fresh water is poured into the tank at a rate of $\backslash (2\backslash$mathrm$\{$gal$\}/\backslash$mathrm$\{$min$\}\backslash ),$ with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional $10$ min $.$