$\backslash$table$[[$ITEM$,1,2,3,4,5,6,7,8],[$WEIGHT$,8,1,7,6,3,12,5,14],[$UTILITY$,80,20,50,55,50,75,30,70]]$

Recognizing that the hike to the campsite is a long one, a limit of $35$ pounds has been set as the maximum total weight of the items to be carried.

$($a$)$ Formulate this as a $0-1$ programming problem to maximize the total utility of the items carried. Solve this knapsack problem using a computer.

$($b$)$ Suppose item $3$ is an extra battery pack, which may be used with several of the other items. Tina has decided that she will take item $5,$ a CD player, only if she also takes item $3.$ On the other hand, if she takes item $3,$ she may or may not take item $5.$ Modify the problem to reflect this, and solve the new problem.