Revenue = (# of Calculators ) * ( price ) Revenue = x(p) R(p) = (21,000 - 150p)p = 21,000p – 150p2 Set each factor equal to zero. 0 < p < 140 or [ 0, 140 ] 21,000 – 150p= 0 p = 0 21,000 = 150p p = 140 Graph it and maximize it. p = $70 R(70) = (21,000 – 150(70))70 = $735,000 x = 21,000 – 150(70) = 10,500 calculators Set R(p) = 675,000 and solve for p. 675,000 = 21,000p – 150p2 0 =( p – 50 )( p – 90 ) Divide out -150 from both sides. - 4500 = p2 – 140p p = 50 or p = 90 0 = p2 – 140p + 4500 50 < p < 90

A farmer has 2000 yards of fence to build two adjacent rectangular corrals. What are the dimensions of each corral to maximize the total area? x x Draw a diagram of two adjacent rectangles. We will need the area formula of a rectangle and label all sides of the rectangles. y y y Area = Base * Height = Length * Width x x 2x y We need an equation for the total of the fencing. 4x + 3y = 2000 Solve for y. 3y = 2000 – 4x y = 2000 – 4x 3

A farmer has 2000 yards of fence to build two adjacent corrals A farmer has 2000 yards of fence to build two adjacent corrals. What are the dimensions of each corral to maximize the total area? Set the function equal to zero and solve for x. This is your domain limits. x x x =250 x =250 0 < x < 500 y y y x x Graph it and MAX IT! y = 2000 – 4(250) 3 = 2000 – 1000 3 = 1000 3 Each corral is 250 yd by 333.3 yd.

positive above below

Greater than zero, means we want positive y-coordinates. ( 0, -12 ) Now find the x-coordinates that generate positive y-coordinates. smallest largest (- , ] U [ , + ) We just need to find the x-intercepts. x = or x = 6 6 -2 -2

( , ) Solve the inequality and graph the solution set. 5 4 2 Set the right side to zero. a = 2, positive, so opens up. y-intercept is ( 0, -10 ). Less than zero, means we want negative y-coordinates. ( 0, -10 ) Now find the x-coordinates that generate negative y-coordinates. smallest largest ( , ) We just need to find the x-intercepts. Big Small 5 4 2

( - , ) Solve the inequality and graph the solution set. a = 1, positive, so opens up. y-intercept is ( 0, 1 ). Greater than zero, means we want positive y-coordinates. We just don’t know which graph we will have. ( 0, 1 ) If we find the x-intercepts, then we will know which graph we have. Complete the square or quadratic formula. Not Possible. No x-intercepts! ( )2 = negatives Therefore, we have the red graph. The entire graph is above the x-axis. ( - , )

Review patterns and sign rules. Must have all terms to the left and zero on the right. Any greater than symbol. The intervals will always be a Union. Any less than symbol. The intervals will always be a single interval. If the leading coefficient, a, is negative, then we must divide both sides by negative one. Remember this will cause the inequality symbol to FLIP!