$1)$ If the profit on a trick ski is $$50$ and the profit on a slalom ski is $$30,$ how many of each type of ski should be manufactured each day to realize a maximum profit? What is the maximum profit?

$2)$ Discuss the effect of the production schedule and the maximum profit if the profit on a slalom ski decreases to $$20?$

$3)$ Discuss the effect of the production schedule and the maximum profit if the profit on a slalom ski increases to $$40.$