SECTION 3.4 Building Quadratic Models from Verbal Descriptions and from Data 171 In this section, we first discuss models that lead to a quadratic function from verbal descriptions.Then we fit a quadratic function to data, which is another form of modeling. When a quadratic function models a problem, the properties of the graph of the function can provide important information about the model. In particular, we can determine the maximum or minimum value of the function. The fact that the graph of a quadratic function has a maximum or minimum value enables us to answer questions involving optimization —that is, finding the maximum or minimum values in models. 1 Build Quadratic Models from Verbal Descriptions In economics, revenue R, in dollars, is defined as the amount of money received from the sale of an item and is equal to the unit selling price p, in dollars, of the item times the number x of units actually sold. That is, R xp = The Law of Demand states that p and x are inversely related: As one increases, the other decreases. The equation that relates p and x is called a demand equation . When a demand equation is linear, the revenue model is a quadratic function. OBJECTIVES 1 Build Quadratic Models from Verbal Descriptions (p. 171) 2 Build Quadratic Models from Data (p. 174) 3.4 Building Quadratic Models from Verbal Descriptions and from Data Now Work the ‘Are You Prepared?’ problems on page 176. • Problem Solving (Section A.8, pp. A66–A73) • Building Linear Models from Data (Section 3.2, pp. 149–153) PREPARING FOR THIS SECTION Before getting started, review the following: Maximizing Revenue The marketing department at Texas Instruments has found that when certain calculators are sold at a price of p dollars per unit, the number x of calculators sold is given by the demand equation x p 21,000 150 = − (a) Find a model that expresses the revenue R as a function of the price p. (b) What is the domain of R? Assume revenue is nonnegative. (c) What unit price should be used to maximize revenue? (d) If this price is charged, what is the maximum revenue? (e) How many units are sold at this price? (f) Graph R. (g) What price should Texas Instruments charge to collect at least $675,000 in revenue? EXAMPLE 1 Solution (a) The revenue is R xp, = where x p 21,000 150 . = − R xp p p p p 21,000 150 150 21,000 2 ( ) = = − = − + The Model (b) Because x represents the number of calculators sold, we have x 0, ≥ so p 21,000 150 0. − ≥ Solving this linear inequality gives p 140. ≤ Also from R p p 21,000 150 , ( ) = − since R is assumed to be nonnegative and p 21,000 150 0, − ≥ it follows that p 0. ≥ Combining these inequalities gives the domain of R, which is p p 0 140 . { } ≤ ≤ (continued)
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