Pre-Calculus
Friday, February 10
Bellwork:
Write a polynomial inequality representing the width of a uniform cement strip around a rectangular
swimming pool of dimensions 30ft by 40ft if enough material is purchased to cover 500 ft 2 of area.
Check and go over homework
Polynomial Inequalities in Two Variables/ Linear Programming
Example #1
Sketch the graph of y < x3 + 2x2
1) Find the zeros.
x2 (x + 2)
x2 = 0
x = 0 (double root)
x+2=0
x = -2
2) Test Pts.
-3: (+)(-) = - (Below x axis)
-1: (+)(+) = + (Above x axis)
1: (+)(+) = + (Above x axis)
THE SHADED REGION IS
BELOW THE CURVE
3) Shading
< means Solid curve
Shade below the curve
Example #2
Graph the solution set of the system
y > -x
y < (x + 2)2
2
y > -x (Linear)
y > (x + 2)
(Quadratic)
b=0
m = -1
line will be dotted
shade above line
vertex: (-2, 0)
y-int: (0,4)
parabola will be solid
shade inside the parabola
The line is
shaded above
This is the region where the
two shadings overlap so it represents
the solution region
Example #3
This region represents where
the two shadings overlap.
Graph the solution of | x - 3| < 2
y> x2 – 4x + 4
1 2
x – 3 < 2 and x – 3 > -2
x < 5 and x > 1
two dotted vertical lines
shade in between lines
5
2
y > x – 4x + 4
y > (x – 2)2
vertex: (2, 0)
y – int (0, 4)
solid parabola
shade inside parabola
Linear Programming
Linear Programming: A mathematical method of finding the maximum profit or minimum cost under a
variety of linear constraints to solve decision-making problems.
Constraints: Sets of linear inequalities given as restrictions on a linear programming problem.
Feasible Region: The region resulting from the intersection of the set of linear inequalities. It contains
all feasible solutions (solutions that satisfies all the requirements of the problems.)
Example
A factory produces short-sleeved and long-sleeved shirts. A short-sleeved shirt requires 30 minutes of
labor, a long-sleeved shirt requires 45 minutes of labor, and 240 hours of labor are available per day.
The maximum number of shirts that can be packaged in a day is 400, so no more than 400 shirts should
be produced. If the profits on a short-sleeved and a long-sleeved shirt are $11 and $16, respectively,
find the maximum possible daily profit.
Short –sleeved: x
Long-sleeved: y
Profit Function: P(x) = 11x + 16y (Don’t Graph This!)
1) x > 0
y>0
2) ½ x + ¾ y < 240 (480,0) (0,320)
400
320
(240, 160)
400 480
3) x + y < 400 (400,0) (0, 400)
P(0,0) = 0 P(0, 320) = 11(0) + 16(320) = 5120
P(400,0) = 11(400) + 16(0) = 4400 P(240, 160) = 11(240) + 16(160)= 5200 (Max)
Therefore there are 240 short sleeved and 160 long sleeved.
Homework: P. 106 #6, 18, 22, 24, 30; p. 114 # 11a