Answer to Question #129631 in Linear Algebra for martha jakob

Question

A company manufactures two types of wedding dresses, mermaid and ballroom style for sale. The
mermaid makes a profit of N$1000 and ballroom makes a profit of N$850. The manufacture of each
of the dresses involves 3 parts, designing, cutting and sewing. The mermaid requires 12hours of
designing, 9 hours of cutting and 30 hours of sewing. The ballroom requires 24hours of designing, 5
hours of cutting and 30 hours of sewing. Due to the number of employees available at the company
there is at most 480 hour available for designing, 180 hours for cutting and 720 hours for sewing.
Assume that the company can sell all the wedding dresses it manufactures, find the maximum number
of each type of wedding dress the company should manufacture to maximize profit.

Answer 1

$$Solution$$

a. ) Choose the unknowns.

$x=Represent\ mermaid\ style$

$y=Represent\ ballroom\ style$

b. ) Write the objective function

$f(x,y)=1000x+850y$

c. ) Write the constraints as a syste

$12x+24y \leq 480\\ 9x+5y \leq 180\\ 30x+30y \leq 720$

As the time taken in wedding dresses are natural numbers, there are two more constraints

x ≥ 0

y ≥ 0

d. ) Find the set of feasible solutions that graphically represent the constraints.

Represent the constraints graphically.

As x ≥ 0 and y ≥ 0, work in the first quadrant.

Solve the inequation graphically, taking a point on the plane, for example

The area of intersection of the solutions of the inequalities would be the solution to the system of inequalities, which is the set of feasible solutions.

e. ) Calculate the coordinates of the vertices from the compound of feasible solutions.

The optimal solution, if unique, is a vertex. These are the solutions to systems:

$12x+24y = 480\ ;\ x=0,\ (0,20) \\ 9x+5y = 180\ ;\ y=0,\ (20,0)\\ 12x+24y = 480,\ 30x+30y = 720,\ (8,16) \\ 30x+30y = 720,\ 12x+24y = 480,\ (15,9)$

f. ) Calculate the value of the objective function at each of the vertices to determine which of them has the maximum or minimum values.

In the objective function, place each of the vertices that were determined in the previous step.

$f(x,y)=1000x+850y$

$f(0,20)=1000(0)+850(20)=17000$

$f(20,0)=1000(20)+850(0)=20000$

$f(8,16)=1000(8)+850(16)=21600$

$f(15,9)=1000(15)+850(9)=22650$

To maximize profit the company should manufacture x=15, y=9.

$f(15,9)=1000(15)+850(9)=22650------>Answer$