Resource Constraints:

Cutting time available: $40$ hours $($or $2400$ minutes$)$

Sewing time available: $52$ hours $($or $3120$ minutes$)$

Production Times:

Mountain Parka:

Cutting time: $30$ minutes

Sewing time: $45$ minutes

Rocky Parka:

Cutting time: $45$ minutes

Sewing time: $15$ minutes

Retail Prices:

Mountain Parka: $$105$

Rocky Parka: $$85$

Minimum Requirement:

At least $20$ Mountain Parkas must be manufactured.

Let$$s denote the number of Mountain Parkas produced as $($x$)$ and the number of Rocky Parkas produced as $($y$).$

Our objective is to maximize the total profit contribution, which can be expressed as:

$[\backslash$text$\{$Total Profit$\}=105$x $+85$y $]$

Subject to the following constraints:

Cutting time constraint: $[30$x $+45$y $\backslash$leq $2400]$

Sewing time constraint: $[45$x $+15$y $\backslash$leq $3120]$

Minimum requirement for Mountain Parkas: $[$ x $\backslash$geq $20]$

Non$-$negativity constraints: $[$ x $\backslash$geq $0,\backslash$quad y $\backslash$geq $0]$

Now, let$$s solve this linear program to find the optimal solution.

Graphical Solution:

We can graph the feasible region defined by the constraints.

The corner points of the feasible region will help us determine the optimal solution.

Corner Points:

Let$$s calculate the coordinates of the corner points:

Point A: $(20,0)$

Point B: $(40,0)$

Point C: $(40,24)$

Point D: $(20,48)$

Evaluate Objective Function at Corner Points:

At Point A: Total Profit $=(105\backslash$cdot $20+85\backslash$cdot $0=2100)$

At Point B: Total Profit $=(105\backslash$cdot $40+85\backslash$cdot $0=4200)$

At Point C: Total Profit $=(105\backslash$cdot $40+85\backslash$cdot $24=5640)$

At Point D: Total Profit $=(105\backslash$cdot $20+85\backslash$cdot $48=4920)$

Optimal Solution:

The maximum total profit occurs at Point C: $($x $=40,$ y $=24).$

Therefore, Expedition Outfitters should manufacture $40$ Mountain Parkas and $24$ Rocky Parkas to maximize the total profit contribution.