$-$ Please study the "duality theory" in reading material: Hillier, F$.$S$.$ and G$.$J$.$ Lieberman, Introduction to Operations Research, $9$th edition, McGraw$-$Hill Science$/$Engineering$/$Math $($Chapter $6).$

Table $3$: Food

Problem $11($bonus$,10$pts$)$ Consider Table $3$ of food and its nutritional values:

The ideal intake for an adult is at least $25$ grams of protein, $40$ grams of carbohydrates, and $400$ calories per day. The problem is to find the least costly way to achieve those amounts of nutrition by using the four types of food shown in the table.

$($a$).$ Formulate this problem as a linear programming problem.

$($b$).$ Formulate the dual problem.