Mat 540 Case Study

Question 1

i) Unknowns x = number of lamps L1 y = number of lamps L2

ii) Function f(x, y) = 15x + 10y

iii) Conversion of time from minutes to hours
20 min = 1/3 h
30 min = 1/2 h
10 min = 1/6 h L1 L2 Time(h)
Manual 1/3 1/2 100
Machine 1/3 1/6 80

iv) Constraints as a system of inequalities
1/3x + 1/2y ≤ 100
1/3x + 1/6y ≤ 80 x ≥ 0 y ≥ 0
v) Solutions that graphically represent the constraints

As x ≥ 0 and y ≥ 0, it lies on the first quadrant
1/3 x + 1/2 y ≤ 100
1/3 • 0 + 1/2 • 0 ≤ 100
1/3 • 0 + 1/6 • 0 ≤ 80
The area of intersection would be the solution to the system of inequalities.

vi) Coordinates of the vertices
1/3x + 1/2y = 100; x = 0 (0, 200)
1/3x + 1/6y = 80; y = 0(240, 0)
1/3x + 1/2y = 100; 1/3x + 1/6y = 80(210, 60) vii)

f(x, y) = 15x + 10y f(0, 200) = 15•0 + 10•200 = $2,000 f(240, 0 ) = 15•240 + 10•0 = $3,600 f(210, 60) = 15•210 + 10•60 = $3,750
The best solution is to manufacture 210 units of model L1 and 60 units of model L2 to obtain the maximum benefit of