Question

It takes Fena Tailoring 3 hr of cutting and 6 hr of sewing to make a tiered silk organza bridal dress. It takes 6 hr of cutting and 3 hr of sewing to make a lace sheath bridal dress. The shop has at most 30 hr per week available for cutting and at most 33 hr per week for sewing. The profit is ​$334 on an organza dress and ​$190 on a lace dress. How many of each kind of bridal dress should be made each week in order to maximize​ profit? What is the maximum​ profit?

Answer 1

Solution-

Let x and y represents the number of organza dresses and ​the lace dresses that should be made per week to maximize the profit.

Since the profit is ​$334 on an organza dress and ​$190 on a lace dress.

So, Profit function in dollars is

Z(x, y) = 334x + 190y .....(1)

Now,

Since it takes Fena Tailoring 3 hr of cutting to make a tiered silk organza bridal dress and 6 hr to make a lace sheath bridal dress.The shop has at most 30 hr per week available for cutting.

So, 3x + 6y ≤ 30

Dividing both sides by 3 to get

x + 2y ≤ 10 ....(2)

Similarly it takes Fena Tailoring 6 hr of sewing to make a tiered silk organza bridal dress and 3 hr of sewing to make a lace sheath bridal dress. shop has at most 33 hr per week for sewing.

So, 6x + 3y ≤ 33

Dividing both sides by 3 to get

2x + y ≤ 11 .....(2)

Now, we have two inequalities.Let us consider these inequalities as equation to find the corner points and solving them graphically.

Equation corrosponding to inequality (2) is

x + 2y =10 .....(4)

At x =0 , y =5 and

At y =0, x = 10

Equation corrosponding to inequality (3) is

2x + y = 11 .....(5)

At x =0, y =11 and

At y =0 , x = 5.5

Now, substracting equation (5) from 2 times if equation (4) to get

2(x+2y) -(2x +y) = 2(10) - 11

2x +4y -2x -y = 20 -11

3y =9

Or y =9/3 = 3 .....(6)

Putting the is value in equation (4) to get

x + 2(3) = 10

Or x +6 =10

Or x =10 -6 = 4 ....(7)

So, from equation (6) and (7) the point (4, 3) is the solution of the two equations.

Now, using the above information, the graphical representation of the solution of inequality (2) and (3) is shown below-

From the graph we can observe that the corner points of the solution are (0, 0), (0, 5), (5.5, 0) and (4, 3).

Now, Since the profit function is (from equation 1)

Z(x,y) = 334x + 190y

Let us check , which corner points will result in maximum profit in dollars.

Z(0,0) = 334(0) +190(0) =0

Z(5,0) = 334(5) + 190(0) = 1670 + 0 = $ 1670

Z(0, 5.5) =334(0) + 190(5.5) = 0 + 1045 =$1045

Z(4, 3)= 334(4) +190(3)= 1336 +570 = $ 1906

So, x =4 and y =3 will result in maximum profit.

Hence,

Number of dress of each kind of bridal dress that should be made each week in order to maximize​ profit are

Organza dress = x = 4

Lace dress = y = 3

And

The maximum​ profit is

Z(4,3) = $1906