SECTION 1.5 Lines 47 133. Product Promotion A cereal company finds that the number of people who will buy one of its products in the first month that the product is introduced is linearly related to the amount of money it spends on advertising. If it spends $40,000 on advertising, then 100,000 boxes of cereal will be sold, and if it spends $60,000, then 200,000 boxes will be sold. (a) Write a linear equation that relates the amount A spent on advertising to the number x of boxes the company aims to sell. (b) How much expenditure on advertising is needed to sell 300,000 boxes of cereal? (c) Interpret the slope. 134. The equation − = x y C 2 defines a family of lines, one line for each value of C. On one set of coordinate axes, graph the members of the family when = − = C C 4, 0, and = C 2. Can you draw a conclusion from the graph about each member of the family? 135. Challenge Problem Find three numbers a for which the lines + = − + = x y x y 2 5,2 3 4 0,and + = ax y 0 do not form a triangle. 136. Challenge Problem Show that the line containing the points ( ) a b , and ( ) b a , , ≠ a b, is perpendicular to the line = y x.Also show that the midpoint of ( ) a b , and ( ) b a , lies on the line = y x. 137. Challenge Problem Form a triangle using the points ( ) ( ) a 0,0, ,0,and ( ) b c , , where > > a b 0, 0, and > c 0. Find the point of intersection of the three lines joining the midpoint of a side of the triangle to the opposite vertex. 138. Challenge Problem Prove that if two nonvertical lines have slopes whose product is −1, then the lines are perpendicular. [Hint: Refer to Figure 69 and use the converse of the Pythagorean Theorem.] 131. Access Ramp A wooden access ramp is being built to reach a platform that sits 30 inches above the floor. The ramp drops 2 inches for every 25-inch run. y Ramp Platform 30 in. x (a) Write a linear equation that relates the height y of the ramp above the floor to the horizontal distance x from the platform. (b) Find and interpret the x-intercept of the graph of your equation. (c) Design requirements stipulate that the maximum run be 30 feet and that the maximum slope be a drop of 1 inch for each 12 inches of run.Will this ramp meet the requirements? Explain. (d) What slopes could be used to obtain the 30-inch rise and still meet design requirements? Source: www.adaptiveaccess.com/wood_ramps.php 132. U.S. Advertising Share A report showed that Internet ads accounted for 19% of all U.S. advertisement spending when print ads (magazines and newspapers) accounted for 26% of the spending. The report further showed that Internet ads accounted for 35% of all advertisement spending when print ads accounted for 16% of the spending. (a) Write a linear equation that relates that percent y of print ad spending to the percent x of Internet ad spending. (b) Find the intercepts of the graph of your equation. (c) Do the intercepts have any meaningful interpretation? (d) Predict the percent of print ad spending if Internet ads account for 39% of all advertisement spending in the United States. Source: Marketing Fact Pack 2018. Ad Age Datacenter 139. Which of the following equations might have the graph shown? (More than one answer is possible.) (a) + = x y 2 3 6 (b) − + = x y 2 3 6 (c) − = − x y 3 4 12 (d) − = x y 1 (e) − = − x y 1 (f) = − y x3 5 (g) = + y x2 3 (h) = − + y x3 3 140. Which of the following equations might have the graph shown? (More than one answer is possible.) (a) + = x y 2 3 6 (b) − = x y 2 3 6 (c) + = x y 3 4 12 (d) − = x y 1 (e) − = − x y 1 (f) = − − y x2 1 (g) = − + y x 1 2 10 (h) = + y x 4 141. The figure shows the graph of two parallel lines.Which of the following pairs of equations might have such a graph? x y y x (a) − = + = x y x y 2 3 2 7 (b) + = + = − x y x y 2 1 (c) − = − − = x y x y 2 1 (d) x y x y 2 2 2 4 − = − − = − (e) + = + = − x y x y 2 2 2 1 Explaining Concepts y x x y 142. The figure shows the graph of two perpendicular lines. Which of the following pairs of equations might have such a graph? (a) − = + = − y x y x 2 2 2 1 (b) − = + = y x y x 2 0 2 0 (c) − = + = − y x y x 2 2 2 2 (d) − = + = − y x x y 2 2 2 1 (e) + = − + = − x y y x 2 2 2 2 143. m is for Slope The accepted symbol used to denote the slope of a line is the letter m. Investigate the origin of this practice. Begin by consulting a French dictionary and looking up the French word monter. Write a brief essay on your findings. 144. Grade of a Road The term grade is used to describe the inclination of a road. How is this term related to the notion of slope of a line? Is a 4% grade very steep? Investigate the grades of some mountainous roads and determine their slopes. Write a brief essay on your findings. Steep 7% Grade
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