Name ——————————————————————— Lesson
3.6
Date ————————————
Challenge Practice
For use with the lesson “Model Direct Variation”
In Exercises 1–4, state whether the two quantities have direct variation. If they do, write an equation that relates the variable quantities and find the constant of variation. 1. Tom is making a rectangular garden. He decides to make the garden 30 feet long
by 15 feet wide. Later, Tom decides to keep the length of the garden the same but change the width. Do the area A and the width W have direct variation? 2. Linda is designing an aquatic tank for a theme park. Because of space
considerations, the bottom of the tank will be square. The area of the bottom of the tank must be 400 square feet and the depth of the tank must be 10 feet in order to provide enough room for the dolphin act to perform. Linda decides to keep the dimensions of the bottom of the tank the same, but change the depth. Under these circumstances do the volume of the tank V and the depth D have direct variation? 3. Suppose in Exercise 1 Tom decides to change both the length and the width of the
garden. Does the area change in direct variation as both the length and width of the garden change? 4. Suppose in Exercise 2 Linda decides to keep the depth of the tank at 10 feet but
change the length of the sides of the bottom of the tank. Does the volume of the tank V vary directly with the side length of the bottom of the tank X ?
cost of 3 tons of sand if the sand costs $1.20 per 100 pounds. (Hint: 1 ton 5 2000 pounds) 6. The number of wolves that can be housed in a wildlife animal sanctuary varies
directly with the number of acres of land available for the wolves to roam. If 100 acres will support 4 wolves, how many acres would be needed to support 10 wolves?
Lesson 3.6
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5. The cost of a truckload of sand varies directly with the weight of the load. Find the
7. For a given space shuttle mission, the cost of adding an additional astronaut to the
mission varies directly with the weight of the astronaut. If it costs $7000 per pound to add an additional astronaut to the mission, what would it cost to add a 170-pound astronaut to the mission? 8. The number of bacteria in a Petri dish varies directly with the area of the dish.
Suppose a dish with an area of 10 square centimeters holds 4000 bacteria. In order to hold 20,000 bacteria what would the area of the dish have to be?
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Lesson 3.6 Model Direct Variation, continued y
Teaching Guide
23
21
1
3 x
23
5 25 3 10. y 5 } x; 6 11. y 5 2} x; 2} 8 4 5 40 4 7 35 12. y 5 } x; } 13. y 5 2} x; 2} 3 3 4 2 10 100 3 14. y 5 } x; } 15. y 5 } x; 3 3 3 10 5 25 2 20 16. y 5 } x; } 17. y 5 2} x; 2} 4 2 7 7 5 1 18. y 5 } x; } 19. yes; y 5 6x 12 6 7 1 20. yes; y 5 2} x 21. no 22. yes; y 5 } x 2 16 1 23. y 5 2x 24. y 5 2} x 25. y 5 25x 5 8 1 26. y 5 } x 27. y 5 23x 28. y 5 } x 3 3 1 9 2 29. y 5 2} x 30. y 5 } x 31. y 5 } x 9 5 2 3 1 1 32. y 5 2} x 33. y 5 } x 34. y 5 2} x 8 8 5 35. no 36. a. Because the ratios for each data pair is 12, s varies directly with n. b. s 5 12n c. 18 lights d. The 200-watt transformer because
2. y 110 100 90 80 70 60 50 6 8 10 12 14 16 18 x Hours worked
y 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0
60 120 180 Distance (miles)
x
The graphs are linear functions because the points plotted form a line. Investigating Algebra Activity
5. No, the equation cannot be written in y 5 ax form.
1. Sample answer: The functions have different slopes, but the same value of b. The graphs of the functions have the same y-intercept, but different (positive) slopes. 2. Sample answer: The functions have the same slope, but different values of b. The graphs of the functions have the same slope, but different y-intercepts; that is, the lines are parallel. 3. Sample answer: The graphs have the same slope (m 5 1), but different y-intercepts. The graphs of this family are vertical translations of the graph of f(x) 5 x. 4. Sample answer: The graphs have different (positive or negative) slopes, but the same y-intercept (0, 0). The graphs of this family are vertical stretches or shrinks (with or without refections in the x-axis) of the graph of f(x) 5 x.
Real-Life Application
Practice A
you need at least a 180-watt transformer. 37. a. s 5 0.07p b. p 5 1.6d c. s 5 0.07(1.6d) ⇒ s 5 0.112d Study Guide 2 3 7 1. yes; } 2. no 3. yes; } 4. y 5 2} x, 28 5 5 9
1. c 5 2.15g 2. $40.85 3. The constant of c variation 4. 2.19 5 }g or 2.19g 5 c 5. $39.42
Challenge Practice 1. Yes, A and W have direct variation. A 5 30W 2. Yes, V and D have direct variation. V 5 400D 3. No, A does not vary directly as both L and W vary. 4. No, V does not vary directly as X varies. 5. $72 6. 250 acres 7. $1,190,000 8. 50 cm2
A34
1. The first table represents a function because for each input there is exactly one output. The second table does not represent a function because there are two ouput values for the input values 13, 14, and 15. The last table represents a function because for each input there is exactly one output.
1. 227; 3; 23 2. 226; 25; 9 3. 7; 4; 2 4. 6; 9; 11 5. 10; 1; 25 6. 215; 23; 5 7. 17; 11; 7 8. 7; 28; 218 9. 23.3; 0; 2.2 1 3 2 10. 9.6; 0; 26.4 11. 21; 0; } 12. }; 0; 2} 2 3 4 3 13. 23 14. 3 15. 3 16. } 17. 7 18. 2 2 19. 2 20. 22 21. 21 22. 4 23. 22 24. 2
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1
Time (hours)
3
Amount earned (dollars)
answers
9.
Lesson 3.7 Graph Linear Functions
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