Question

Maximize p = 3x + 5y + 7z subject to

x + y + z ≤ 150
x + y + z ≥ 100
x ≥ 0, y ≥ 0, z ≥ 0. p=__________
(x,y,z)=________
Minimize c = 18x + 18y subject to
x + 2y ≥ 20
2x + y ≥ 20
x ≥ 0, y ≥ 0. c=__________
(x,y,)=________
Minimize c = 4x + y + 2z subject to
x + y + z ≥ 100
2x + y ≥ 60
y + z ≥ 60
x ≥ 0, y ≥ 0, z ≥ 0. c=_______
(x,y,z)=_____

Answer 1

1. Solution: p = 675, (x,y,z) = (0,75,75).

Explanation: To maximize p = 3x + 5y + 7z subject to the given constraints, we can find the value of z from the first constraint: z = 150 - x - y.

Then, substitute this into the objective function and differentiate with respect to x and y to find the maximum point.
Solution: p = 675, (x,y,z) = (0,75,75).

2. Solution: c = 360, (x,y) = (10,5).

Explanation: To minimize c = 18x + 18y subject to the given constraints, you can solve the system of linear inequalities graphically or algebraically.

In this case, we can use algebra to find the corner points and then evaluate the objective function at those points.
Solution: c = 360, (x,y) = (10,5).

3. Solution: c = 220, (x,y,z) = (20,40,40).

Explanation: To minimize c = 4x + y + 2z subject to the given constraints, we can use the graphical method or the simplex method to find the optimal solution.
Solution: c = 220, (x,y,z) = (20,40,40).