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 the objective function

The profit function $P$ to be maximized is $P = 130x+120y$.

Step3: Set up the constraints

For assembly - labor: $1.2x + 0.6y\leq600$. For packaging - labor: $0.5x+1y\leq400$. Also, $x\geq0,y\geq0$ since the number of units cannot be negative. Rewrite the assembly - labor constraint as $y\leq - 2x + 1000$ and the packaging - labor constraint as $y\leq - 0.5x + 400$.

Step4: Find the intersection points of the constraints

First, find the intersection of $y=-2x + 1000$ and $y=-0.5x + 400$. 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 $y=-0.5\times400 + 400=200$. The intersection of $y=-2x + 1000$ with the $x$ - axis is when $y = 0$, so $0=-2x+1000$, $x = 500$. The intersection of $y=-0.5x + 400$ with the $x$ - axis is when $y = 0$, so $0=-0.5x + 400$, $x = 800$. The intersection of $y=-2x + 1000$ with the $y$ - axis is $(0,1000)$ and the intersection of $y=-0.5x + 400$ with the $y$ - axis is $(0,400)$.

Step5: Evaluate the objective function at the corner - points

Corner - point 1: $(0,0)$, $P(0,0)=130\times0+120\times0 = 0$. Corner - point 2: $(0,400)$, $P(0,400)=130\times0+120\times400=48000$. Corner - point 3: $(400,200)$, $P(400,200)=130\times400+120\times200=52000 + 24000=76000$. Corner - point 4: $(500,0)$, $P(500,0)=130\times500+120\times0=65000$.

Answer:

400 grill A units and 200 grill B units.