Question

Given the linear programming problem, use the method of corners to determine where the minimum occurs and give the minimum value.

Minimize

Exam Image

Subject to
x ≤ 3
y ≤ 9
x + y ≥ 9
x ≥ 0
y ≥ 0

Answer 1

Answer:

Minimum value of function $C=x+10y$ is 63 occurs at point (3,6).

Step-by-step explanation:

To minimize :

                                   $C=x+10y$

Subject to constraints:

                                   $x\leq 3---(1)\\y\leq 9---(2)\\x+y\geq 9----(3)\\x\geq 0\\y\geq 0$

Eq (1) is in blue in figure attached and region satisfying (1) is on left of blue line

Eq (2) is in green in figure attached and region satisfying (2) is below the green line

Considering $x+y\geq 9$, corresponding coordinates point to draw line are (0,9) and (9,0).

Eq (3) makes line in orange in figure attached and region satisfying (3) is above the orange line

Feasible region is in triangle ABC with common points A(0,9), B(3,9) and C(3,6)

Now calculate the value of function to be minimized at each of these points.

                                    $C=x+10y$

at A(0,9)

                                     $C=0+10(9)\\C=90$

at B(3,9)

                                     $C=3+10(9)\\C=93$

at C(3,6)

                                     $C=3+10(6)\\C=63$

Minimum value of function $C=x+10y$ is 63 occurs at point C (3,6).