$[$Dynamic Programming $1](16$ points$)$

There are $n$ sites for a franchise along a long road at positions $(x_1,$dots,$x_n)$ and you can assume that $r_{i}5kmF[k]1,$dots,$k{x}_1.$

There $is$ a rule $in$ the franchise handbook that $itis$ forbidden for two franchises can $be$ within $5$ miles $of$ each other.

Each franchise site has a projected revenue $r_i.$

You wish $to$ determine the maximum amount $of$ total projected revenue $of$ a set $of$ franchise sites you can open that follow the constraints that two franchises cannot $be$ within $5km$ apart.

Step $1$: Description $of$ subproblems

Let $F[k]be$ the maximum projected revenue from the franchises $1,$dots,$k.$

Step $2$: Base Case$(s)(2$ points$)$

Step $3$: Recursion $(with$ justification $or$ case analysis$)(8$ points$)$

Step $4$: ordering $(2$ point$)$

Step $5$: form $of$ output $(2$ point$)$

Step $6$: runtime analysis $(2$ points$)$