Find the minimum of the objective function, f, subject to the constraints. f=7x+4y, constraints= 5x+y≥60, x+6y≥60, 4x+6y≥204, x≥0, y≥0. thanks!!

Answer 1

f = 7x + 4y at a minimum is found by plugging in 5 points, the intersections of the boundaries of the 5 constraints

(0,60) gives 240

(6,30) f= 42+120 = 162

(10.3,8.3) = 72.1+ 33.2 = 105.3

(48,2) = 336+ 8 = 344

(60,0) = 420

minimum f is 105 at the point about (10.3,8,3)

that's the intersection of 5x+y = 60 and x+6y =60

multiply 1st by 6 and subtract the 2nd equation

30x + 6y = 360

x + 6y = 60

29x = 300

x = 300/29 = about 10.345 rounded to 10.3

y = 60-5x = 60 - 5(10.345) = 60-51.724 = 8.276 rounded to 8.3

(10.3, 8.3) minimizes the function f = 7x+4y