Question

By graphing the system of constraints find the values of x and y that maximize the objective function, find the maximum value.

x+y<=11
2y>=x
x>=0
y>=0

Maximum for P=2x+y

A. P=10 1/3
B. P=7 1/3
C. P=21
D. P=29 1/3

Answer 1

First, find the vertices of the shaded region.

x≥0, y≥0 represents the first quadrant.

x + y ≤ 11 has intercepts at (11,0) and (0, 11)

2y ≥ x has intercepts at (0, 0)

x + y ≤ 11 and 2y ≥ x intercept at
`((22)/(3), (22)/(6) )`

The vertices within the shaded region are (0, 11), (0, 0), and
`((22)/(3), (22)/(6) )`.

Next, evaluate the function P = 2x + y at those intercepts.

P = 2x + y at (0, 11) ⇒ P = 2(0) + (11) ⇒ P = 0 + 11 ⇒ P = 11

P = 2x + y at (0, 0) ⇒ P = 2(0) + (0) ⇒ P = 0 + 0 ⇒ P = 0

P = 2x + y at
`((22)/(3), (22)/(6) )` ⇒ P = 2
`((22)/(3))` +
`((22)/(6) )` ⇒ P =
`(44)/(3)` +
`(22)/(6)` ⇒ P =
`(110)/(6)` ⇒ P =
`(55)/(3)` = 19
`(1)/(3)`

The maximum is found at
`((22)/(3), (22)/(6) )`

Answer: maximum is 19
`(1)/(3)`