32 Chapter 1 FUNCTIONS AND ALGEBRAIC STRUCTURE 10. y = x+2 Exercises for Section 1.5 3x IDENTIFYING ALGEBRAIC STRUCTURE 9. y = 1 12. y = 9x - 5x For the formula in Exercises 1-14, is y directly proportional to x? If so, give the constant of proportionality. 11. y = 3(x +2) 14. y - 5 = 2x 1. y = 5x 2. y = x . 7 13. y = (3x) 2 3. y = -x 4. y = x . x 5. y = V5. x 6. y = V5x 7. y = x/9 8. y = 9/x PROBLEMS 27. 26. y In Problems 15-18 is the first quantity proportional to the second quantity? If so, what is the constant of proportional- NHOHN ity? N 15. d is the distance traveled in miles and t is the time trav- eled in hours at a speed of 50 mph. X -2 -1 01 16. P is the price paid in dollars for b barrels of oil at a price 2 1 2 3 4 of $70. 29 17. p is the sale price of an item whose original price is Po 28. 5 in a 30% off sale. O 18. C is the cost of having n drinks at a club, where each drink is $5 and there is a cover charge of $20. X - 5 In Problems 19-22, assume the two quantities are directly proportional to each other. 10 3 19. If r = 36 when s = 4, find r when s is 5. 31. y 20. If p = 24 when q = 6, find q when p is 32. 30. 2 21. If y = 16 when x = 12, find y when x is 9. O 22. If s = 35 when t = 25, find t when s is 14. 23. The interest paid by a savings account in one year is - 2 proportional to the starting balance, with constant of -3 proportionality 0.06. Write a formula for I, the amount 32. y of interest earned, in terms of B, the starting balance. Find the interest earned if the starting balance is -NW . $500 (b) $1000 (c) $5000. In Problems 24-32, determine if the graph defines a direct 2 3 proportion. If it does, estimate the constant of proportional- ity. 33. A factory makes 50 vehicles a week. Is t, the number of 24. 25. y weeks, proportional to u, the number of vehicles made? 20 If so, what is the constant of proportionality? 15 10 34. One store sells 70 pounds of apples a week, and a sec- ond store sells 50 pounds of apples a week. Is the total 3 6 9 12 2 4 6 8 10 number of pounds of apples sold, a, proportional to the number of weeks, w? If so, what is the constant of pro- portionality?32 Chapter 1 FUNCTIONS AND ALGEBRAIC STRUCTURE 10. y = x+2 Exercises for Section 1.5 3x IDENTIFYING ALGEBRAIC STRUCTURE 9. y = 1 12. y = 9x - 5x For the formula in Exercises 1-14, is y directly proportional to x? If so, give the constant of proportionality. 11. y = 3(x +2) 14. y - 5 = 2x 1. y = 5x 2. y = x . 7 13. y = (3x) 2 3. y = -x 4. y = x . x 5. y = V5. x 6. y = V5x 7. y = x/9 8. y = 9/x PROBLEMS 27. 26. y In Problems 15-18 is the first quantity proportional to the second quantity? If so, what is the constant of proportional- NHOHN ity? N 15. d is the distance traveled in miles and t is the time trav- eled in hours at a speed of 50 mph. X -2 -1 01 16. P is the price paid in dollars for b barrels of oil at a price 2 1 2 3 4 of $70. 29 17. p is the sale price of an item whose original price is Po 28. 5 in a 30% off sale. O 18. C is the cost of having n drinks at a club, where each drink is $5 and there is a cover charge of $20. X - 5 In Problems 19-22, assume the two quantities are directly proportional to each other. 10 3 19. If r = 36 when s = 4, find r when s is 5. 31. y 20. If p = 24 when q = 6, find q when p is 32. 30. 2 21. If y = 16 when x = 12, find y when x is 9. O 22. If s = 35 when t = 25, find t when s is 14. 23. The interest paid by a savings account in one year is - 2 proportional to the starting balance, with constant of -3 proportionality 0.06. Write a formula for I, the amount 32. y of interest earned, in terms of B, the starting balance. Find the interest earned if the starting balance is -NW . $500 (b) $1000 (c) $5000. In Problems 24-32, determine if the graph defines a direct 2 3 proportion. If it does, estimate the constant of proportional- ity. 33. A factory makes 50 vehicles a week. Is t, the number of 24. 25. y weeks, proportional to u, the number of vehicles made? 20 If so, what is the constant of proportionality? 15 10 34. One store sells 70 pounds of apples a week, and a sec- ond store sells 50 pounds of apples a week. Is the total 3 6 9 12 2 4 6 8 10 number of pounds of apples sold, a, proportional to the number of weeks, w? If so, what is the constant of pro- portionality?1.5 FUNCTIONS, MODELING, AND PROPORTIONALITY 33 35. The perimeter of a square is proportional to the length (a) Write a formula expressing C as a function of x. of any side. What is the constant of proportionality? b) One type of denim costs $28.50 for 3 yards. Find 36. A bike shop's revenue is directly proportional to the the constant of proportionality and give its units. number of bicycles sold. When 50 bicycles are sold, the (c) How much does 5.5 yards of this type of denim cost? revenue is $20,000. (a) What is the constant of proportionality, and what 42. The distance M, in inches, between two points on a map are its units? is proportional to the actual distance d, in miles, be- b) What is the revenue if 75 are sold? tween the two corresponding locations. 37. The total cost of purchasing gasoline for your car (a) If 1/2 inch represents 5 miles, find the constant of is directly proportional to the the number of gallons proportionality and give its units. pumped, and 11 gallons cost $36.63. (b) Write a formula expression M as a function of d. (a) What is the constant of proportionality, and what c) How far apart are two towns if the distance between them on the map is 3.25 inches? are its units? (b) How much do 15 gallons cost? 43. The blood mass of a mammal is proportional to its body 38. The required cooling capacity, in BTUs, for a room air mass. A rhinoceros with body mass 3000 kilograms has conditioner is proportional to the area of the room be- blood mass of 150 kilograms. Find a formula for the ing cooled. A room of 280 square feet requires an air blood mass of a mammal in terms of the body mass and conditioner whose cooling capacity is 5600 BTUs. estimate the blood mass of a human with body mass 70 kilograms. (a) What is the constant of proportionality, and what are its units? 44. A bond pays 5% interest per year. If you buy $1000 b) If an air conditioner has a cooling capacity of of the bond, you receive $50 per year. For $1500 of 10,000 BTUs, how large a room can it cool? the bond, you receive $75/yr, for $5000 of the bond, 39. You deposit $P into a bank where it earns 2% interest $250/yr. What is the average rate of change of interest per year for 10 years. Use the formula B = P(1 + r)' per dollar change of bond purchased: for the balance $B, where r is the interest rate (written (a) Between $1000 and $1500? as a decimal) and t is time in years. b) Between $1500 and $5000? (a) Explain why B is proportional to P. What is the constant of proportionality? 45. The data rate of an Internet connection is the rate in (b) Is P proportional to B? If so, what is the constant bytes per second that data, such as a web page, im- of proportionality? age, or music file, can be transmitted across the connec- tion. Suppose the data rate is 300 kilobytes per second 40. The length m, in inches, of a model train is proportional KBps). How long does it take to download a file that to the length r, in inches, of the corresponding real train. is 60,000 kilobytes (KB)? (a) Write a formula expressing m as a function of r. 46. The distance a car travels on the highway is proportional (b) An HO train is 1/87th the size of a real train. 13 to the quantity of gas consumed. A car travels 225 miles What is the constant of proportionality? What is on 5 gallons of gas. Find the constant of proportionality, the length in feet of a real TGV train if the HO train give units for it, and explain its meaning. is 90.62 inches long? (c) A Z scale train is 1/220th the size of a real train. 47. If x is directly proportional to y, explain what happens What is the constant of proportionality? What is to y if x is the length, in inches, of a Z scale Eurostar train if the real Eurostar is 1293 feet long?14 (a) doubled. (b) halved. 41. The cost of denim fabric is directly proportional to the (c) multiplied by a factor k. amount that you buy. Let C be the cost, in dollars, of x yards of denim fabric. (d) multiplied by a factor 1 K ' 13 www.internettrains.com, accessed September 1, 2014. 14 Train sizes from: en. wikipedia.org/wiki/TGV#Rolling_stock, accessed September 1, 2014. 15 Sometimes mistakenly called "bandwidth." See the Free Online Dictionary of Computing, http://foldoc.doc.ic.ac.uk/foldoc/contents.html, accessed September 1, 2014.1.5 FUNCTIONS, MODELING, AND PROPORTIONALITY 33 35. The perimeter of a square is proportional to the length (a) Write a formula expressing C as a function of x. of any side. What is the constant of proportionality? b) One type of denim costs $28.50 for 3 yards. Find 36. A bike shop's revenue is directly proportional to the the constant of proportionality and give its units. number of bicycles sold. When 50 bicycles are sold, the (c) How much does 5.5 yards of this type of denim cost? revenue is $20,000. (a) What is the constant of proportionality, and what 42. The distance M, in inches, between two points on a map are its units? is proportional to the actual distance d, in miles, be- b) What is the revenue if 75 are sold? tween the two corresponding locations. 37. The total cost of purchasing gasoline for your car (a) If 1/2 inch represents 5 miles, find the constant of is directly proportional to the the number of gallons proportionality and give its units. pumped, and 11 gallons cost $36.63. (b) Write a formula expression M as a function of d. (a) What is the constant of proportionality, and what c) How far apart are two towns if the distance between them on the map is 3.25 inches? are its units? (b) How much do 15 gallons cost? 43. The blood mass of a mammal is proportional to its body 38. The required cooling capacity, in BTUs, for a room air mass. A rhinoceros with body mass 3000 kilograms has conditioner is proportional to the area of the room be- blood mass of 150 kilograms. Find a formula for the ing cooled. A room of 280 square feet requires an air blood mass of a mammal in terms of the body mass and conditioner whose cooling capacity is 5600 BTUs. estimate the blood mass of a human with body mass 70 kilograms. (a) What is the constant of proportionality, and what are its units? 44. A bond pays 5% interest per year. If you buy $1000 b) If an air conditioner has a cooling capacity of of the bond, you receive $50 per year. For $1500 of 10,000 BTUs, how large a room can it cool? the bond, you receive $75/yr, for $5000 of the bond, 39. You deposit $P into a bank where it earns 2% interest $250/yr. What is the average rate of change of interest per year for 10 years. Use the formula B = P(1 + r)' per dollar change of bond purchased: for the balance $B, where r is the interest rate (written (a) Between $1000 and $1500? as a decimal) and t is time in years. b) Between $1500 and $5000? (a) Explain why B is proportional to P. What is the constant of proportionality? 45. The data rate of an Internet connection is the rate in (b) Is P proportional to B? If so, what is the constant bytes per second that data, such as a web page, im- of proportionality? age, or music file, can be transmitted across the connec- tion. Suppose the data rate is 300 kilobytes per second 40. The length m, in inches, of a model train is proportional KBps). How long does it take to download a file that to the length r, in inches, of the corresponding real train. is 60,000 kilobytes (KB)? (a) Write a formula expressing m as a function of r. 46. The distance a car travels on the highway is proportional (b) An HO train is 1/87th the size of a real train. 13 to the quantity of gas consumed. A car travels 225 miles What is the constant of proportionality? What is on 5 gallons of gas. Find the constant of proportionality, the length in feet of a real TGV train if the HO train give units for it, and explain its meaning. is 90.62 inches long? (c) A Z scale train is 1/220th the size of a real train. 47. If x is directly proportional to y, explain what happens What is the constant of proportionality? What is to y if x is the length, in inches, of a Z scale Eurostar train if the real Eurostar is 1293 feet long?14 (a) doubled. (b) halved. 41. The cost of denim fabric is directly proportional to the (c) multiplied by a factor k. amount that you buy. Let C be the cost, in dollars, of x yards of denim fabric. (d) multiplied by a factor 1 K ' 13 www.internettrains.com, accessed September 1, 2014. 14 Train sizes from: en. wikipedia.org/wiki/TGV#Rolling_stock, accessed September 1, 2014. 15 Sometimes mistakenly called "bandwidth." See the Free Online Dictionary of Computing, http://foldoc.doc.ic.ac.uk/foldoc/contents.html, accessed September 1, 2014