Math 28 Project I Introduction To start, consider the last time you purchased something, such as cereal. Let's say you bought 5 boxes of cereal at $3.00. Now, what if there was a sale? You might be more likely to buy more than 5. However, if there was a price increase, you would probably buy less. Most people would buy less as the price increases 12 10 8 Quantity bought 6 4 2 0 $- $1.00 $2.00 $3.00 $4.00 $5.00 $6.00 Price (in dollars) Price function For each price level of a product, there is a corresponding quantity of that product that consumers will demand (purchase) during some time period. Usually, the higher the price, the smaller is the quantity demanded; as the price falls, the quantity demanded increases. If the price per unit of the product is given by p and the corresponding quantity is given by q, then an equation relating p and q is called a demand equation. We'll make a simplifying assumption that the demand equation is linear. A good question for thinking deeper is how you address more complex types of equations, such as piecewise linear. Since negative prices or quantities are not meaningful, both p and q are both nonnegative. If we solve for p in the demand equation, the resulting equation is called a price function. For most products, an increase in the quantity demanded corresponds to a decrease in price. Thus, a typical price curve falls from left to right. Below is an example of a price curve that goes through the points (0, 1000) and (100, 800): Figure 1 - Example price function. Price decreases as quantity sold increases. 1200 1000 1000 800 800 600 Price (in dollars) 400 200 0 0 20 40 60 80 100 120 Quantity Sold In fact, we see that the price function is the line through the points (0, 1000) and (100, 800): 1000 800 p1000 (q 0) 0 100 , which simplifies to p 2q1000 . Recall that revenue is defined to be the income generated by the sale of goods or services: Revenue=Price Quantity= pq So given our price function p(q)=2 q+ 1000 , the revenue function R(q) is R ( q ) =pq=(2 q+1000 ) q=2q 2+ 1000 q . Note that R is a quadratic function of q, with a=2 , b = 1000, and c = 0. Since a < 0, the graph of R is a parabola that opens downward, so R has a maximum at the b b ,R vertex : 2a 2a ( ( )) b 1000 q 250 2a 2(2) . Thus the maximum value of R is given by R 2(250)2 1000(250) 125, 000. This says that the maximum revenue that the manufacturer can make is $125,000, which occurs at a production level of 250 units. Assignment: Based on the above discussion, complete the problem below, showing your work in the spaces provided. Through a series of marketing pilot programs, the marketing department determines the demand function for a small computer company's premier laptop is p q= +400 , where p is the price (in dollars) per unit when q units are 6 demanded (per week) by consumers. Find the level of production per week that will maximize total revenue, and determine the maximum revenue per week. a) Solve for p to determine the price function p(q): b) Using the fact that Revenue = price x quantity, determine the revenue function R(q): c) Find the q-coordinate of the vertex, for the parabola that corresponds to R(q). This is the level of production per week that will maximize total revenue: d) Find the R-coordinate of the vertex, for the parabola that corresponds to R(q). This is the maximum revenue per week