7. 0 / 2 points details my notes previous answers ask your teacher practice another a division of the…

7. 0 / 2 points details my notes previous answers ask your teacher practice another a division of the winston furniture company manufactures dining tables and chairs. each table requires 40 board feet of wood and 3 labor - hours. each chair requires 16 board feet of wood and 4 labor - hours. the profit for each table is $45, and the profit for each chair is $30. in a certain week, the company has 3200 board feet of wood available and 520 labor - hours available. how many tables and chairs should winston manufacture to maximize its profit? (let x represent the number of tables winton manufactures and let y represent the number of chairs they manufacture.) (x, y)=(80,0)× what is the maximum profit? $3600× resources ebook

7. 0 / 2 points details my notes previous answers ask your teacher practice another a division of the winston furniture company manufactures dining tables and chairs. each table requires 40 board feet of wood and 3 labor - hours. each chair requires 16 board feet of wood and 4 labor - hours. the profit for each table is $45, and the profit for each chair is $30. in a certain week, the company has 3200 board feet of wood available and 520 labor - hours available. how many tables and chairs should winston manufacture to maximize its profit? (let x represent the number of tables winton manufactures and let y represent the number of chairs they manufacture.) (x, y)=(80,0)× what is the maximum profit? $3600× resources ebook

Answer

Explanation:

Step1: Set up constraints

Wood - constraint: $40x + 16y\leq3200$, simplifies to $5x + 2y\leq400$. Labor - constraint: $3x+4y\leq520$. Also, $x\geq0,y\geq0$.

Step2: Rewrite inequalities as equations for boundary - lines

$5x + 2y=400$ gives $y = 200-\frac{5}{2}x$. $3x + 4y=520$ gives $y = 130-\frac{3}{4}x$.

Step3: Find intersection points

Solve the system of equations $\begin{cases}5x + 2y=400\3x + 4y=520\end{cases}$. Multiply the first equation by 2: $10x+4y = 800$. Subtract the second equation: $(10x + 4y)-(3x + 4y)=800 - 520$, $7x=280$, $x = 40$. Substitute $x = 40$ into $5x+2y=400$, $200+2y=400$, $y = 100$. Intersection point is $(40,100)$. Also, find intersection with axes: For $5x + 2y=400$, when $x = 0,y = 200$; when $y = 0,x = 80$. For $3x + 4y=520$, when $x = 0,y = 130$; when $y = 0,x=\frac{520}{3}\approx173.33$.

Step4: Set up profit function

The profit function is $P=45x + 30y$.

Step5: Evaluate profit function at corner - points

Corner - points are $(0,0),(0,130),(80,0),(40,100)$. $P(0,0)=45\times0+30\times0 = 0$. $P(0,130)=45\times0+30\times130=3900$. $P(80,0)=45\times80+30\times0 = 3600$. $P(40,100)=45\times40+30\times100=1800 + 3000=4800$.

Answer:

The number of tables $x = 40$ and chairs $y = 100$ should be manufactured. The maximum profit is $$4800$.