Learning Curve The management at a certain factory has found that a worker can produce at most $30$ units in a day. The learning curve for the number of units $N$ produced per day after a new employee has worked $t$ days is

$N=30(1-e^{kt}).$

After $20$ days on the job, a particular worker produces $19$ units.

$($a$)$ Find the learning curve for this worker.

$($b$)$ How many days should pass before this worker is producing $25$ units per day?