Linear Patterns and Contexts Quiz

PDF page 16: Since the unit rate of feet to yards is 3, the unit rate of yards to feet is 1/3.

PDF page 16: There are 5280 feet in a mile. How many yards are in a mile? 1 yard 5280 = yards = 1760 yards.

PDF page 17: In basketball, it is necessary to have 12 players in a roster. In a particular district in Eastern Utah, the middle school basketball league has teams that are made up of boys and girls. For fairness, it is decided that each team must have 7 girls and 5 boys. This tells us that the ratio of girls to boys in the basketball league is 7:5. The relation, girls to boys in the basketball league is a proportional relationship, with constant of proportionality “girls to boys” equal to 7/5.

PDF page 17: Question 2. There are 45 boys eligible for basketball. How many girls are needed to to complete the league? Here we want to think in terms of the constant of proportionality, which is 7/5. So the number

PDF page 17: of girls needed is 7/5 of the number of boys available; that is, (7/5) × 45 = 7 × 9 = 63.

PDF page 17: Question 1. If Gramps drives 5 hours, how far does he go? Here, we think of unit rate: the rate of miles per hour is 30. Since miles × hours , miles = hours he traveled 30 × 5 = 150 miles. Question 2: Gramps wants to drive to St. George from Logan; that is 440 miles. How long will it take him at that rate. Here we want to convert to minutes, and the concept of ratio: the ration of minutes to miles is 2:1. So to drive 440 miles, takes Gramps 880 minutes, or 880/60 = 14. 6667, or 14 hours and 40 minutes.

PDF page 18: Notice that, we have accounted for the weight of the container itself by first measuring it empty, and then subtracting that weight from the weight of the container and water at each measurement. We now graph the these data, plotting height along the horizontal axis and weight on the vertical: The graph appears to be a straight line, giving confirmation of our hypothesis that the height of the column of water and its weight are proportional. We can calculate the unit rate of change using any one of the measurements: for example, 4 inches of the water weighs 49.6 ounces, so we have 12.4 ounces per inch of water. This is expressed by the relation W = 12.4H. In an actual experiment there always will be slight variations due errors or estimation, given the accuracy of the instruments used. So, the rates computed from each measurement may differ slightly. We will return to this in the statistics chapter.

PDF page 18: Notice that, we have accounted for the weight of the container itself by first measuring it empty, and then subtracting that weight from the weight of the container and water at each measurement. We now graph the these data, plotting height along the horizontal axis and weight on the vertical: The graph appears to be a straight line, giving confirmation of our hypothesis that the height of the column of water and its weight are proportional. We can calculate the unit rate of change using any one of the measurements: for example, 4 inches of the water weighs 49.6 ounces, so we have 12.4 ounces per inch of water. This is expressed by the relation W = 12.4H. In an actual experiment there always will be slight variations due errors or estimation, given the accuracy of the instruments used. So, the rates computed from each measurement may differ slightly. We will return to this in the statistics chapter.

PDF page 19: The graph of these data appears to be a straight line through the origin suggesting a proportional relationship: Notice that whenever the value of x doubles, so does the value of y, and that a change in x of 1 unit is accompanied by a change in y of 1.8 units. Finally, when we calculate the quotient y/x for any pair of points, we always get the value 1.8. This can be phrased this way: the proportional relationship y = 1.8x models the given data.

PDF page 19: • If quantities y and x are in proportion then the graph of pairs (x, y) in this relation will be a straight line

PDF page 19: through the origin. That line is characterized by the assertion that y/x is constant, and in fact, is the constant of proportionality. In terms of the graph, we call this its slope.

PDF page 19: The graph (see the figure above) is a straight line that does not go through the origin: even if the salesman

PDF page 20: sells no cars, he receives the base salary of $1000. Also note that to each increment of 4 cars sold, the salesman receives an increase of $1000. In particular we can say that the increase in income is to the increase of number of sales as 1000:4 giving us a unit rate of $250 in compensation per unit of cars sold. This is just the coefficient of N in the equation C = 1000 + 250N. To restate this: the number 250 expresses a relation between the variables N and C, even though the variables are not proportional. It is the change in C that is proportional to the change in N at the ratio 250:1.

PDF page 20: At the statewide championship game, each player on each team receives five complimentary tickets, and can buy additional tickets at $20 each. Carlos wants 8 tickets and Louis wants 16 tickets. How much does each pay for the full set of tickets? Solution. One might to say that, since Louis is getting twice as many tickets, he has to pay twice as much. But that would be a mistake, the cost is not proportional to the number of tickets, but cost is proportional to the number of tickets in excess of 5 . In this situation, they each get 5 complimentary tickets, so Carlos pays for 3 tickets and Louis pays for 11 tickets. At $20 apiece, Carlos pays $60 and Louis pays $220.

PDF page 20: By applying this thinking to the general case, we can write down a formula for the cost C of N tickets for any player. If a player wants N tickets, he gets 5 free and pays $20 each for the remaining tickets. There are N − 5 remaining, so the cost is C = 20(N − 5) or C = 20N − 100. The form of these equations tell different things, both interesting. The first ( C = 20(N − 5) ) tells us that the cost is proportional to the excess of tickets above the first 5. The second tells us that the cost is $20 per ticket, less $100 for the free 5 tickets. Note that these equations make sense only for N ≥ 5; players don’t get refunded if they have less than 5 friends. In figure 2 we have graphed this relationship: 350 300 250 200 150 Tickets 100 50 0 10 20 5 15 25 Cost

PDF page 20: At the statewide championship game, each player on each team receives five complimentary tickets, and can buy additional tickets at $20 each. Carlos wants 8 tickets and Louis wants 16 tickets. How much does each pay for the full set of tickets? Solution. One might to say that, since Louis is getting twice as many tickets, he has to pay twice as much. But that would be a mistake, the cost is not proportional to the number of tickets, but cost is proportional to the number of tickets in excess of 5 . In this situation, they each get 5 complimentary tickets, so Carlos pays for 3 tickets and Louis pays for 11 tickets. At $20 apiece, Carlos pays $60 and Louis pays $220.