Finding Maximums Minimums of Polynomial Functions fguilbert Objective
*********************** Finding Maximums & Minimums of Polynomial Functions fguilbert
*********************** Objective - find the maxim and minimum value for a function. fguilbert
*********************** The maximum or minimu value of a quadratic is at the vertex. x= -b 2 a fguilbert
*********************** Ex 1 ***** A rectangular dog p constructed with 60 m of fen One side of the pen is a bar What dimensions will give th greatest area. fguilbert
*********************** 60 m of fence. ***** One side of the pen is a b Find greatest area. barn pen fence fguilbert
*********************** Total of 3 sides is 60. barn X pen 60 - 2 X fguilbert X
*********************** Area = x ( 60 - 2 x) =- 2 2 x + 60 x barn X pen 60 - 2 X fguilbert X
*********************** Area = x ( 60 - 2 x) =- 2 2 x + 60 x vertex (15, 450) fguilbert
*********************** Area = x ( 60 - 2 x) (15, 450) =- 2 2 x + barn 15 fguilbert pen 30 15 60 x
*********************** ex 2 6 x 10 sheet of metal. ***** Squares with side x are cut fr corners. A box is made. Find x to maximize volume. x 6 fguilbert x 10
*********************** A cubic function has a loca max and local min. fguilbert
*********************** 10 ***** x 6 x fguilbert x x x Visualize cutting corners & folding the box.
*********************** 10 - 2 x ***** 6 -2 x fguilbert x x x x Sides of the box are shorter than 10 by 6.
*********************** bottom 10 - 2 x fguilbert 6 -2 x
*********************** x bottom 10 - 2 x 6 -2 x What is the height ? fguilbert
*********************** Volume = l w h x bottom 10 - 2 x 6 -2 x V = (10 - 2 x )( 6 - 2 x ) x fguilbert
*********************** V = (10 - 2 x )( 6 - 2 x max when x = 1. 21 V = 32. 8 fguilbert
*********************** Ex A ball is thrown up. ***** Height is a function of time Given : ho = 15 m 15 vo= 20 m/sec 2 g = 9. 8 m/s h = height fguilbert
*********************** h 0 = 15 Vo =20 g =9. 8 ***** h = - 1/2 g + vot + ho 2 h = - 1/2 (9. 8) t + 20 t + 15 2 h = - 4. 9 t + 20 t + 15 fguilbert 2 t
*********************** b. Find ball height when t = 3 ***** 2 h = - 4. 9 t + 20 t + 15 t t 2 h = -4. 9( 3) 3 + 20(3) + 15 h = 30. 9 meters fguilbert
*********************** When is the ball 25 m off ***** ground? Ball is up 25 M so let h = 25 2 h = - 4. 9 t + 20 t + 15 2 + 25 = - 4. 9 t 20 t + 15 fguilbert
*********************** 2 + 25 = - 4. 9 t 20 t + 15 Put in standard form Get 0 on left. 2 t 0 = - 4. 9 + 20 t - 10 Solve with quadratic formula. fguilbert
*********************** d. When will ball hit ground? ***** Looking for t? Or h? ? h=0 when the ball hits ground 0 = - 4. 9 fguilbert 2 + t 20 t + 15
*********************** 2 + 0 = - 4. 9 t 20 t + 15 Use quadratic formula t = 4. 68 or t = -. 6 (is this ok? ) fguilbert
*********************** When will the ball be at ***** the highest point? Find the maximum height. h = - 4. 9 fguilbert 2 t + 20 t + 15
*********************** I know how to get maximum heigh Just graph the equation. Find th y value of the m fguilbert
*********************** Find the Max ( t, h) (2. 04, 35. 41) h = - 4. 9 fguilbert 2 t + 20 t +15
*********************** Effort It is a solo performance that only you can control. " fguilbert
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