Question

PLZ HELP ASAP

Answer 1

Answer:

You cannot find the error because the graph was not included, but you can find the correct solution.

The correct solution is:

• The maximum of P is  9

The graph is attached and the resultant region is the quadrilateral formed by the vertices (0,0), (0,3), (1,4), and (3,0).

Explanation:

To find the maximum value of P = - x + 3y subject to the constraints you should draw the graph showing the region determined by the constraints, and then use the the coordinates of the vertices of the bounded region to find which one yields the maximum value of P.

1. Graph

Constrain               Step                                          shade

y ≤ -2 x + 6            draw solid line y = - 2x + 6        region below the line

y ≤ x + 3                 draw solid line y = x + 3            region below the line

x ≥ 0, and y ≥ 0                                                        first quadrant.

2. How to graph the lines

• To graph the line y = -2x + 6, you choose two points:

         x = 0 ⇒ y = -2(0) + 6 = 6 ⇒ (0,6)

         y = 0 ⇒ 0 = -2x + 6 ⇒ 2x = 6 ⇒ x = 3 ⇒ (3,0)

• To graph the line y = x + 3, you choose two points:

         x = 0 ⇒ y = 0 + 3 = 3 ⇒ (0,3)

         y = 0 ⇒ 0 = x + 3 ⇒ x = - 3 ⇒ ( -3, 0)

3. Vertices

The graph shows which intersection points you must find.

a) Intersection of the positive axis: origin (0,0)

b) Intersection of y = 0 and y = -2x + 6:

   0 = - 2x + 6 ⇒ 2x = 6 ⇒ x = 3 ⇒ point (3,0)

c) Intersection of x = 0 and y = x + 3

   y = 0 + 3 ⇒ y = 3 ⇒ point (0,3)

d) Intersection of y = -2x + 6 and y = x + 3

    - 2x + 6 = x + 3 ⇒ 3 = 3x ⇒ x = 1

     y = 1 + 3 = 4

     ⇒ point (1, 4)

Summarizing, the intersection points are: (0,0), (3,0), (0,3), and (1,4)

The graph is attached and the resultant region is the quadrilateral formed by the vertices (0,0), (0,3), (1,4), and (3,0).

4.  Maximization

The maximum and minimum values of the function are in the vertices, so you must find the value of the function P for every vertex of the bounded region.

This table shows you how you do it and the results:

Point     x     y         P = - x + 3y

(0,0)      0    0              - 0 + 0 = 0

(0,3)      0     3             - 0 + 3(3) = 9    ←  maximum

(3,0)      3     0             - 3 + 0 = - 3      ← minimum

(1, 4)      1      4              - 1 + 4 = 3

Thus, the maximum value is 9.