Material Taken From: Mathematics for the international student Mathematical Studies SL Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark Bruce Haese and Haese Publications, 2004

Section 9L - Quadratic Modeling Remember: The vertex is the maximum and minimum value of a parabola. Therefore, in word problems, if we are going to find the maximum area, profit, etc, then we simply have to find the vertex.

For y = ax 2 + bx + c Vertex is: If a> 0 then: If a< 0 then:

The height H meters, of a rocket t seconds after it is fired vertically upwards is given by H(t) = 80t – 5t 2, t  0. a)How long does it take for the rocket to reach its maximum height? a)What is the maximum height reached by the rocket? a)How long does it take for the rocket to fall back to earth? Example Number 1

A vegetable gardener has 40 meters of fencing to enclose a rectangular garden plot where one side is an existing brick wall. If the two equal sides are x meters long: a)Show that the area enclosed is given by A = x(40 – 2x) m 2 a)Find the dimensions of the vegetable garden of maximum area. Example Number 2

A fence encloses a rectangular yard on three sides. The fence has a total length of 56 meters. a)Write down expressions for the width and length of the yard, in terms of x. a)Find the maximum possible area of the yard, and the corresponding dimensions. Example Number 3

Example Number 4

The daily profit P (in Euros) made by a business depends on the number of workers employed. The daily profit is modeled by the function P(x) = -50x x – 2000, where x is the number of workers employed on any given day. a)Using the axes provided below, sketch the graph of P(x) = -50x x – b)Determine the number of workers required to maximize the profit. c)Find the maximum possible profit. d)A flu virus prevents all but 1 worker attending work. Calculate the maximum amount of money the business losses on that day. Example Number 5

A company manufactures and sells CD-players. If x CD-players are made and sold each week, the weekly cost $C to the company is C(x) = x The weekly income $I = 50x. a) Determine an expression for the weekly profit P(x) given P(x) = I(x) – C(x) if x CD-players are produced and sold. b) Determine the number of CD-players which must be made and sold each week to gain the maximum profit. Example Number 6

A small manufacturing company makes and sells x machines each month. The monthly cost C, in dollars, of making x machines is given by C(x) = x 2. The monthly income I, in dollars, obtained by selling x machines is given by I(x) = 150x – 0.6x 2. (a) Show that the company’s monthly profit can be calculated using the quadratic function P(x) = – x x – (b) The maximum profit occurs at the vertex of the function P(x). How many machines should be made and sold each month for a maximum profit? (c) If the company does maximize profit, what is the selling price of each machine? (d) Given that P(x) = (x – 20) (130 – x), find the smallest number of machines the company must make and sell each month in order to make positive profit. Example Number 7

Homework Exercise 9L, pg 310 – #1 through 8