144 CHAPTER 3 Linear and Quadratic Functions NOTE Even though the linear function V has a domain of all real numbers, in this application the implied domain of ( ) V x is ≤ ≤x 0 7. j (e) To find when the book value will be $9000, solve the equation V x x x x 9000 4500 31,500 9000 4500 22,500 22,500 4500 5 ( ) = − + = − = − = − − = The car will have a book value of $9000 when it is 5 years old. Subtract 31,500 from both sides. Divide by −4500. Now Work PROBLEM 47 Supply and Demand The quantity supplied of a good is the amount of a product that a company is willing to make available for sale at a given price. The quantity demanded of a good is the amount of a product that consumers are willing to purchase at a given price p. Suppose that the quantity supplied, S, and the quantity demanded, D, of smartphones each month are given by the following functions: S p p D p p 15 900 3.75 2850 ( ) ( ) = − = − + where p is the price (in dollars) of the smartphone. (a) The equilibrium price of a product is defined as the price at which quantity supplied equals quantity demanded. That is, the equilibrium price is the price at which S p D p . ( ) ( ) = Find the equilibrium price of smartphones.What is the equilibrium quantity, the amount demanded (or supplied) at the equilibrium price? (b) Determine the prices for which quantity supplied is greater than quantity demanded. That is, solve the inequality S p D p . ( ) ( ) > (c) Graph S S p( ) = and D D p , ( ) = and label the equilibrium point, the point of intersection of S and D. Solution EXAMPLE 5 (a) To find the equilibrium price, solve the equation S p D p . ( ) ( ) = p p p p p p 15 900 3.75 2850 15 3.75 3750 18.75 3750 200 − = − + = − + = = The equilibrium price is $200 per smartphone. To find the equilibrium quantity, evaluate either S p( ) or D p( ) at p 200. = S 200 15 200 900 2100 ( ) = ⋅ − = The equilibrium quantity is 2100 smartphones. At a price of $200 per phone, the company will produce and sell 2100 phones each month and have no shortages or excess inventory. (b) The inequality S p D p ( ) ( ) > is p p p p p p 15 900 3.75 2850 15 3.75 3750 18.75 3750 200 − >− + >− + > > If the company charges more than $200 per phone, quantity supplied will exceed quantity demanded. In this case the company will have excess phones in inventory. ( ) = − S p p 15 900; ( ) =− + D p p 3.75 2850 Add 900 to both sides. Add p 3.75 to both sides. Divide both sides by 18.75. ( ) ( ) > S p D p Add 900 to both sides. Add p 3.75 to both sides. Divide both sides by 18.75.
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