6. -/2 points kane manufacturing has a division that produces two models of fireplace grates, x units of…

6. -/2 points kane manufacturing has a division that produces two models of fireplace grates, x units of model a and y units of model b. to produce each model a grate requires 3 lb of cast iron and 6 min of labor. to produce each model b grate requires 4 lb of cast iron and 3 min of labor. the profit for each model a grate is $2.50, and the profit for each model b grate is $2.50. also, 1000 lb of cast iron and 20 labor - hours are available for the production of fireplace grates per day. because of a backlog of orders for model a grates, kanes manager had decided to produce at least 150 of these grates a day. operating under this additional constraint, how many grates of each model should kane produce to maximize profit? (x,y) = ( ) what is the optimal profit? $ resources ebook

6. -/2 points kane manufacturing has a division that produces two models of fireplace grates, x units of model a and y units of model b. to produce each model a grate requires 3 lb of cast iron and 6 min of labor. to produce each model b grate requires 4 lb of cast iron and 3 min of labor. the profit for each model a grate is $2.50, and the profit for each model b grate is $2.50. also, 1000 lb of cast iron and 20 labor - hours are available for the production of fireplace grates per day. because of a backlog of orders for model a grates, kanes manager had decided to produce at least 150 of these grates a day. operating under this additional constraint, how many grates of each model should kane produce to maximize profit? (x,y) = ( ) what is the optimal profit? $ resources ebook

Answer

Explanation:

Step1: Set up the constraints

The cast - iron constraint: $3x + 4y\leq1000$. The labor - time constraint: $6x+3y\leq20\times60 = 1200$. The minimum production of model A: $x\geq150$. Also, $x\geq0,y\geq0$.

Step2: Set up the objective function

The profit function $P = 2.5x+2.5y$.

Step3: Rewrite the inequalities as equations for boundary - line analysis

For the cast - iron constraint $3x + 4y=1000$, we can express $y=\frac{1000 - 3x}{4}$. For the labor - time constraint $6x + 3y=1200$, we can express $y = 400 - 2x$.

Step4: Consider the intersection of constraints

First, consider the intersection of $x = 150$ and $3x+4y=1000$. Substitute $x = 150$ into $3x+4y=1000$, we get $3\times150+4y=1000$, $450+4y=1000$, $4y=550$, $y=\frac{550}{4}=137.5$. Next, consider the intersection of $x = 150$ and $6x + 3y=1200$. Substitute $x = 150$ into $6x+3y=1200$, we have $6\times150+3y=1200$, $900+3y=1200$, $3y = 300$, $y = 100$. We also need to find the intersection of $3x + 4y=1000$ and $6x+3y=1200$. Multiply the first equation by 2: $6x+8y=2000$. Subtract the second equation $6x + 3y=1200$ from it: $(6x+8y)-(6x + 3y)=2000 - 1200$, $5y=800$, $y = 160$. Then substitute $y = 160$ into $3x+4y=1000$, $3x+4\times160=1000$, $3x+640=1000$, $3x=360$, $x = 120$ (but this point does not satisfy $x\geq150$).

Step5: Evaluate the profit function at corner - points

Corner - point 1: $(x = 150,y = 100)$ $P=2.5\times150+2.5\times100=2.5\times(150 + 100)=2.5\times250=625$. Corner - point 2: $(x = 150,y = 137.5)$ $P=2.5\times150+2.5\times137.5=2.5\times(150 + 137.5)=2.5\times287.5 = 718.75$.

Answer:

$(x,y)=(150,137.5)$ The optimal profit is $$718.75$