a manufacturer produces two models of a gas grill. grill a requires 1.2 hr for assembly and 0.5 hr for…

a manufacturer produces two models of a gas grill. grill a requires 1.2 hr for assembly and 0.5 hr for packaging. grill b requires 0.6 hr for assembly and 1 hr for packaging. the production information and profit for each grill are given in the table. the manufacturer has 600 hr of labor available for assembly and 400 hr of labor available for packaging. (a) determine the number of grill a units and the number of grill b units that should be produced to maximize profit assuming that all grills will be sold. the manufacturer should produce grill a units and grill b units to maximize profit. part: 0 / 3 part 1 of 3

a manufacturer produces two models of a gas grill. grill a requires 1.2 hr for assembly and 0.5 hr for packaging. grill b requires 0.6 hr for assembly and 1 hr for packaging. the production information and profit for each grill are given in the table. the manufacturer has 600 hr of labor available for assembly and 400 hr of labor available for packaging. (a) determine the number of grill a units and the number of grill b units that should be produced to maximize profit assuming that all grills will be sold. the manufacturer should produce grill a units and grill b units to maximize profit. part: 0 / 3 part 1 of 3

Answer

Explanation:

Step1: Define variables

Let $x$ be the number of grill A units and $y$ be the number of grill B units.

Step2: Set up constraints

Assembly - time constraint: $1.2x + 0.6y\leq600$. Packaging - time constraint: $0.5x+1y\leq400$. Also, $x\geq0,y\geq0$ (non - negative production).

Step3: Set up the objective function

The profit function $P = 110x + 80y$.

Step4: Rewrite the inequalities as equations for boundary lines

Assembly: $1.2x + 0.6y=600$ simplifies to $y=-2x + 1000$. Packaging: $0.5x + y=400$ simplifies to $y=-0.5x + 400$.

Step5: Find the intersection point of the boundary lines

Set $-2x + 1000=-0.5x + 400$. $-2x+0.5x=400 - 1000$. $-1.5x=-600$. $x = 400$. Substitute $x = 400$ into $y=-0.5x + 400$, we get $y=-0.5\times400+400=200$.

Step6: Evaluate the objective function at the corner points

Corner point 1: $(0,0)$, $P(0,0)=110\times0 + 80\times0=0$. Corner point 2: Intersection of $x = 0$ and $1.2x+0.6y = 600$ (i.e., $y = 1000$), $P(0,1000)=110\times0+80\times1000 = 80000$. Corner point 3: Intersection of $y = 0$ and $0.5x + y=400$ (i.e., $x = 800$), $P(800,0)=110\times800+80\times0=88000$. Corner point 4: $(400,200)$, $P(400,200)=110\times400+80\times200=44000 + 16000=60000$.

Answer:

400 grill A units and 200 grill B units.