360 CHAPTER 6 Algebra, Graphs, and Functions Did You Know? How to Succeed in Business Economics, a science dependent on mathematics, dates back to just before the Industrial Revolution of the eighteenth century. Technologies were being invented and applied to the manufacture of cloth, iron, transportation, and agriculture. These new technologies led to the development of mathematically based economic models that often included systems of equations. French economist Jules Dupuit (1804–1866) suggested a method to calculate the value of railroad bridges; Irish economist Dionysis Larder (1793–1859) showed railroad companies how to structure their rates so as to increase their profits. b) The break-even point is the point at which the revenue and cost graphs intersect. In Fig. 6.29, the graphs intersect at the point (10, 350), which is the break-even point. Thus, for Richard to break even, he must sell 10 model trains. When 10 model trains are sold, the cost and revenue are both $350. Number of Model Trains Cost and Revenue ($) 2 4 6 8 1012141618 350 100 50 150 200 250 300 400 500 450 y x (10, 350) C 5 200 1 15x R 5 35x Figure 6.29 c) Profit is equal to the revenue minus the cost. Therefore, the profit formula is P R C x x x x x 35 (200 15 ) 35 200 15 20 200 = − = − + = − − = − For 15 trains, the profit is determined as follows. = − = − = P x 20 200 20(15) 200 100 Richard has a profit of $100 if he sells 15 model trains. By observing the graph, we can see that if Richard sells 15 model trains, he will have a profit since at 15 trains the revenue line is above the cost line. d) We can determine the number of model trains that Richard must sell to have a profit of $600 by using the profit formula. Substituting 600 for P we have P x x x x 20 200 600 20 200 800 20 40 = − = − = = Thus, Richard must sell 40 model trains to make a profit of $600. 7 Now try Exercise 51 Example 10 A Mixture Problem Shandra, a pharmacist, needs 500 milliliters (m ) of a 10% phenobarbital solution. She has only a 5% phenobarbital solution and a 25% phenobarbital solution available. How many milliliters of each solution should she mix to obtain the desired solution? Solution First we set up a system of equations. The unknown quantities are the amount of the 5% solution and the amount of the 25% solution that must be used. Library of Congress (LC-DIG-pga-00601) Wavebreakmedia/Shutterstock
RkJQdWJsaXNoZXIy NjM5ODQ=