Name Date Teacher Practice A Direct Variation The following tables show direct variation for the given equation. Complete the missing information in the tables. 1. y = 2x 2. y = 1 3 x x 10 7 3 15 22 y 8 12 24 x 21 9 3 8 19 y 5 4 10 3. Determine whether the data sets show direct variation. x y 8 4 6 3 0 0 2 1 4 2 6 3 Find each equation of direct variation, given that y varies directly with x. 4. y is 10 when x is 2. 5. y is 42 when x is 6. 6. y is 50 when x is 5. 7. y is 15 when x is 30. 8. At a constant speed, the gasoline a car uses varies directly with the distance the car travels. A car uses 10 gallons of gasoline to travel 210 miles. How many gallons will the car use to travel 294 miles?
Name Date Teacher Practice B Direct Variation Determine whether the data sets show direct variation. 1. x y 6 9 4 6 0 0 2 3 8 12 2.Write the equation of direct variation for Exercise 1. Find each equation of direct variation, given that y varies with x. 3. y is 32 when x is 4 4. y is 10 when x is 20 _ 5. y is 63 when x is 7 6. y is 40 when x is 50 _ 7. y is 87.5 when x is 25 8. y is 90 when x is 270 9.The table shows the length and width of various U.S. flags. Determine whether there is direct variation between the two data sets. If so, find the equation of direct variation. _ Length (ft) 2.85 5.7 7.6 9.88 11.4 Width (ft) 1.5 3 4 5.2 6
Name Date Teacher LESSON 12-5 CODE Practice C Direct Variation Find each equation of direct variation, given that y varies directly with x. 1. y is 189 when x is 45 2. y is 456 when x is 3800 3. y is 763 when x is 981 4. y is 171 3 4 when x is 916 Tell whether each equation represents direct variation between x and y. 5. y = 9 10 x 6. y = xy 8 7. 5x y = 0 8. y = 24 x 9. y x = 8.25 10. x y = 10 11. x = y 12. 1 3 y = x 13. The following table shows the distance on a map in inches x and the actual distance between two cities in miles, y. Determine whether there is direct variation between the two data sets. If so, find the equation of direct variation. x 456 1 4 3 1 2 4 5 7 1 4 8 9 1 8 11 y 75 175 200 350 362 1 2 400 456 1 4 550 14. A person s weight on Earth varies directly with a person s estimated weight on Venus. If a person weighs 110 pounds on Earth, he or she would weigh an estimated 99.7 pounds on Venus. If a person weighs 125 pounds on Earth, what would be his or her estimated weight to the nearest tenth of a pound on Venus?
Name Date Teacher Review for Mastery Direct Variation Two data sets have direct variation if they are related by a constant ratio, the constant of proportionality. A graph of the data sets is linear and passes through (0, 0). y = kx equation of direct variation, where k is the constant ratio To determine whether two data sets have direct variation, you can compare ratios. You can also graph the data sets on a coordinate grid. x 3 5 8 y 15 25 40 y x = 15 3 = 25 5 = 40 8 = 5 1 k = 5 y = 5x constant ratio The graph of the data sets is linear and passes through (0, 0). So, the data sets show direct variation. Determine whether the data sets show direct variation. If there is a constant ratio, identify it and write the equation of direct variation. Plot the points and tell whether the graph is linear. 1. 2. x 1 2 4 8 y 8 4 2 1 x 0 2 3 5 y 0 20 30 50 constant ratio? If yes, equation. Is the graph linear? constant ratio? If yes, equation. Is the graph linear?
Name Date Teacher Challenge Different Paths, Same Result Problems of direct variation can be solved with two methods. If r varies directly with h, and r = 13.5 when h = 3, find r when h = 7. Method 1: Find the constant of variation. Method 2: Write a proportion. 13.5 3 r h = k = k Use a pair of known values. 4.5 = k constant of variation r = 4.5h equation of variation r = 4.5(7) = 31.5 So, when h = 7, r = 31.5. 13.5 3 r 1 h 1 = r 2 h 2 = r 2 7 3r 2 = 13.5(7) 3r 2 3 = 94.5 3 r 2 = 31.5 Use all known values. Cross multiply. Use both methods to solve each problem. 1. y varies directly as x. If y = 16 when x = 5, find y when x = 9. _ So, when x = 9, y =. 2. A varies directly as s 2. If A = 75 when s = 5, find A when s = 7. So, when s = 7, A =
Name Date Teacher Reading Strategies Use Tables and Graphs When quantities are related proportionally by a constant multiplier, they have direct variation. This table shows the relationship between the number of glasses filled and the amount of juice needed to fill them. The amount of juice needed varies directly with the number of glasses filled. Glasses 1 2 3 4 Juice Needed 8 oz 16 oz 24 oz 32 oz 1. What are the quantities that form this direct variation? 2. What is the constant multiplier? A graph of a direct variation is always linear and always passes through (0, 0). 3. What do the x-values on the graph stand for? 4. What do the y-values on the graph stand for? 5. What does the ordered pair (2, 16) mean? 6. Write an ordered pair for 3 glasses and the amount of juice needed.
Name Date Teacher Problem Solving Direct Variation Determine whether the data sets show direct variation. If so, find the equation of direct variation. 1. The table shows the distance in feet traveled by a falling object in certain times. 2. The R-value of insulation gives the material s resistance to heat flow. The table shows the R-value for different thicknesses of fiberglass insulation. Time (s) 0 0.5 1 1.5 2 2.5 3 Distance (ft) 0 4 16 36 64 100 144 Thickness (in.) 1 2 3 4 5 6 R-value 3.14 6.28 9.42 12.56 15.7 18.84 3. The table shows the lifting power of hot air. Hot Air (ft 3 ) Lift (lb) 4. The table shows the relationship between degrees Celsius and degrees Fahrenheit. Celsius 10 5 0 5 10 20 30 Fahrenheit 14 23 32 41 50 68 86 Your weight on Earth varies directly with your weights on other planetary bodies. The table below shows how much a person who weighs 100 lb on Earth would weigh on the moon and different planets. 5. Find the equation of direct variation for the weight on earth e and on the moon m. A m = 0.166e C m = 6.02e B m = 16.6e D m = 1660e 6. How much would a 150 lb person weigh on Jupiter? F 63.5 lb H 354.6 lb G 286.4 lb J 483.7 lb Planetary Bodies Weight (lb) Moon 16.6 Jupiter 236.4 Pluto 6.7 7. How much would a 150 lb person weigh on Pluto? A 5.8 lb C 12.3 lb B 10.05 lb D 2238.8 lb
Name Date Teacher Puzzles, Twisters & Teasers It Just Doesn t Hold Water! Circle words from the list in the word search (horizontally, vertically or diagonally). You will also find a word that answers the riddle. direct variation constant proportionality ratio graph quantity algebra table data What is as round as a dishpan and as deep as a tub, yet the ocean could not fill it? A.
2. y = 1.5x or y = 3 x 3. y = 8x 2 Practice A 1. x 10 7 4 3 6 12 15 22 y 20 14 8 6 12 24 30 44 4. y = 1 x 2 5. y = 9x 6. y = 4 x 5 7. y = 3.5x 2. 3. x 21 15 9 3 8 12 19 30 y 7 5 3 1 2 2 3 4 6 1 3 10 8. y = 1 3 x 9. There is direct variation between the lengths and widths of the flags. y = 1.9x, where y is the length, x is the width, and 1.9 is the constant of proportionality The data sets show direct variation. 4. y = 5x 5. y = 7x 6. y = 10x 7. y = 1 2 x 8. 14 gallons Practice B 1. Practice C 1. y = 4.2x 2. y = 0.12x 3. y = 7 9 x 4. y = 3 16 x 5. yes 6. no 7. yes 8. no 9. yes 10 no 11. yes 12. yes 13. There is no direct variation. 14. 113.3 pounds Review for Mastery 1. no no The data sets show direct variation. 2. yes, 10 y = 10x yes
A 1 (s 1 ) 2 = A 2 (s 2 ) 2 75 5 = A 2 ; 2 25A2 = 75 (49) 2 7 25A 2 25 = 3675 25 A 2 = 147; 147 Challenge 1. y x = k 2. 16 5 = k 3.2 = k y = 3.2x y = 3.2(9) = 28.8; y 1 x 1 = y 2 x 2 16 5 = y 2 9 5y 2 = 16(9) y 2 5 = 144 5 y 2 = 28.8; 28.8 A s 2 = k 75 5 2 = k; k = 3 A = 3s 2 A = 3(7 2 ) = 147; Problem Solving 1. No direct variation 2. Direct variation; R = 3.14t 1 3. Direct variation; L = H 50 4. No direct variation 5. A 6. H 7. B Reading Strategies 1. number of glasses filled and ounces of juice needed 2. 8 3. the number glasses filled 4. the amount of juice needed for each glass 5. 2 glasses need 16 ounces of juice 6. (3, 24) Puzzles, Twisters & Teasers sieve