Question

the graph represents the feasible region for the system:
y ≤ 2x
\tx + y ≥ 45
\tx ≤ 30
minimize the objective function p = 25x + 20y.
the minimum value =
and occurs when x =
and y =

Explanation:

Step1: Identify feasible region vertices

The vertices of the feasible region (from the graph and constraints) are:

1. Intersection of $x=30$ and $y=2x$: $(30, 60)$
2. Intersection of $x=30$ and $x+y=45$: $(30, 15)$
3. Intersection of $y=2x$ and $x+y=45$: Solve $x+2x=45 \implies 3x=45 \implies x=15$, $y=30$, so $(15, 30)$

Step2: Evaluate $P=25x+20y$ at each vertex

• For $(30, 60)$:

$P=25(30)+20(60)=750+1200=1950$

• For $(30, 15)$:

$P=25(30)+20(15)=750+300=1050$

• For $(15, 30)$:

$P=25(15)+20(30)=375+600=975$

Step3: Find the minimum value

Compare the calculated $P$ values: $975 < 1050 < 1950$

Answer:

The minimum value = $975$
and occurs when $x = 15$
and $y = 30$