Advanced Mathematical Concepts Extra Examples - DOC - DOC by DOd6Xt70

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									Advanced Mathematical Concepts                                                          Chapter 2

Lesson 2-6

Example 1
CANDY Jennifer is making a party mix using two different kinds of candy, candy corn and
licorice. She wants the total amount of candy in the mix to be no more than five pounds and wants
to use at least 3 pounds of licorice. What combinations of amounts of candy corn and licorice will
satisfy Jennifer’s requirements?

First, write two inequalities that represent each of Jennifer’s requirements. Let c represent the number of
pounds of candy corn and l represent the number of pounds of licorice.

Total:        c+l5
Licorice:     l3

Both of these inequalities include the boundary line, so
the lines are solid. The graph of c + l  5 is composed of
all points on and below the line c + l = 5. The graph of
l  3 includes all points on and above the line l = 3. The
orange area is the solution to the system of inequalities.
That is, the ordered pair for any point in the orange area
satisfies both inequalities.




Example 2

a. Solve the system of inequalities by graphing.
   x2
   y4
   x–y3

b. Name the coordinates of the vertices of the polygonal convex set.

a. Since each inequality contains an equality, the
   boundary lines will be solid. The shaded region
   shows points that satisfy all three inequalities.

b. The region is a triangle whose vertices are the points
   (2, -1), (7, 4), and (2, 4).
Advanced Mathematical Concepts                                                    Chapter 2

Example 3
Find the maximum and minimum values of f(x, y) = 2x + y for the polygonal convex set determined
by the system of inequalities.
    x  -2          x+y8      y1            x–y≤2

Boundary a                Boundary b                  Boundary c        Boundary d
x  -2                    x+y 8                      y1                x-y ≤2
                             y  -x + 8                                   -y ≤ -x + 2
                                                                           y ≥x-2

Graph the inequalities and find the coordinates of the
vertices of the resulting polygon.

The coordinates of the vertices are (-2, 1), (-2, 10),
(5, 3), and (3, 1).

Now, evaluate the function f(x, y) = 2x + y at each
vertex.

f(-2, 1) = 2(-2) + 1 or –3
f(-2, 10) = 2(-2) + 10 or 6
f(5, 3) = 2(5) + 3 or 13
f(3, 1) = 2(3) + 1 or 7

The maximum value of the function is 13, and the minimum value is –3.

								
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