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.\n| | grill a | grill b |\n|--|--|--|\n| assembly | 1.2 hr | 0.6 hr |\n| packaging | 0.5 hr | 1 hr |\n| profit | $110 | $80 |\nthe manufacturer has 600 hr of labor available for assembly and 400 hr of labor available for packaging.\npart 1 of 3\n(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.\nthe manufacturer should produce 400 grill a units and 200 grill b units to maximize profit.\npart 2 of 3\n(b) what is the maximum profit under these constraints?\nthe maximum profit is $

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.\n| | grill a | grill b |\n|--|--|--|\n| assembly | 1.2 hr | 0.6 hr |\n| packaging | 0.5 hr | 1 hr |\n| profit | $110 | $80 |\nthe manufacturer has 600 hr of labor available for assembly and 400 hr of labor available for packaging.\npart 1 of 3\n(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.\nthe manufacturer should produce 400 grill a units and 200 grill b units to maximize profit.\npart 2 of 3\n(b) what is the maximum profit under these constraints?\nthe maximum profit is $

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, $x\geq0,y\geq0$.

Step3: Define the profit function

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

Step4: Solve the system of inequalities

From the assembly - time constraint $1.2x + 0.6y = 600$ can be rewritten as $y=-2x + 1000$. From the packaging - time constraint $0.5x + y=400$ can be rewritten as $y=-0.5x + 400$. 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 $y=-0.5\times400+400=200$.

Step5: Calculate the profit

Substitute $x = 400$ and $y = 200$ into the profit function $P=110\times400+80\times200$. $P = 44000+16000$. $P = 60000$.

Answer:

$60000$