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- 1. Section 6.8 - Variation<br />
- 2. Four types of variation in this section:<br />
- 3. Four types of variation in this section:<br />Direct Variation<br />
- 4. Four types of variation in this section:<br />Direct Variation<br />Inverse Variation<br />
- 5. Four types of variation in this section:<br />Direct Variation<br />Inverse Variation<br />Combined Variation (really just a combination of #1 and #2)<br />
- 6. Four types of variation in this section:<br />Direct Variation<br />Inverse Variation<br />Combined Variation (really just a combination of #1 and #2)<br />Joint Variation<br />
- 7. Direct Variation<br /> <br /> The relationship “y varies directly as x” or “y is proportional to x” could be written in equation form as:<br />
- 8. Direct Variation<br /> <br /> The relationship “y varies directly as x” or “y is proportional to x” could be written in equation form as:<br />y= k x<br />
- 9. Direct Variation<br /> <br /> The relationship “y varies directly as x” or “y is proportional to x” could be written in equation form as:<br /> where k represents the constant of proportionality<br />y= k x<br />
- 10. Direct Variation<br /> <br /> The relationship “y varies directly as x” or “y is proportional to x” could be written in equation form as:<br /> where k represents the constant of proportionality<br />y= k x<br />Example<br /> <br />The circumference,C, of a circle varies directly to the diameter, d, of the circle as seen in the equation: <br />
- 11. Direct Variation<br /> <br /> The relationship “y varies directly as x” or “y is proportional to x” could be written in equation form as:<br /> where k represents the constant of proportionality<br />y= k x<br />Example<br />The circumference,C, of a circle varies directly to the diameter, d, of the circle as seen in the equation: <br />C= π d<br />
- 12. Direct Variation<br /> <br /> The relationship “y varies directly as x” or “y is proportional to x” could be written in equation form as:<br /> where k represents the constant of proportionality<br />y= k x<br />Example<br />The circumference,C, of a circle varies directly to the diameter, d, of the circle as seen in the equation: <br />In this case the constant of proportionality is the number π≈ 3.14<br />C= π d<br />
- 13. So why call it direct variation?<br />
- 14. So why call it direct variation?<br />Look at the circle below. What will happen to the circle’s circumference (distance around) if we make the diameter(distance across) bigger?<br />d = 5 inches<br />C = π(5) ≈ 15.7 in.<br />
- 15. So why call it direct variation?<br />Look at the circle below. What will happen to the circle’s circumference (distance around) if we make the diameter(distance across) bigger?<br />d = 5 inches<br />C = π(5) ≈ 15.7 in.<br />
- 16. So why call it direct variation?<br />Look at the circle below. What will happen to the circle’s circumference (distance around) if we make the diameter(distance across) bigger?<br />d = 8 inches<br />C = π(8) ≈ 25.12 in.<br />d = 5 inches<br />C = π(5) ≈ 15.7 in.<br />
- 17. So why call it direct variation?<br />Look at the circle below. What will happen to the circle’s circumference (distance around) if we make the diameter(distance across) bigger?<br />Notice that increasing the diameter also increased the circumference. This is typical of things that vary directly. Increasing one variable will also increase the other variable. Likewise decreasing one variable will also decrease the other variable. This is why it is called directvariation, whatever you do to one variable (increase or decrease) it will directly affect the other variable in the same way.<br />d = 8 inches<br />C = π(8) ≈ 25.12 in.<br />d = 5 inches<br />C = π(5) ≈ 15.7 in.<br />
- 18. Exercise 1:<br />Write a general equation that represents the given relationship.<br />The electric resistance, R, of a wire varies directly as its length, L.<br />The volume, V, of a sphere varies directly as the cube of its radius.<br />
- 19. Exercise 1:<br />Write a general equation that represents the given relationship.<br />The electric resistance, R, of a wire varies directly as its length, L.<br />The volume, V, of a sphere varies directly as the cube of its radius.<br />
- 20. Exercise 1:<br />Write a general equation that represents the given relationship.<br />The electric resistance, R, of a wire varies directly as its length, L.<br />The volume, V, of a sphere varies directly as the cube of its radius.<br />Remember: direct variation always looks like y = kx. However, we now have different variables.<br />
- 21. Exercise 1:<br />Write a general equation that represents the given relationship.<br />The electric resistance, R, of a wire varies directly as its length, L.<br />The volume, V, of a sphere varies directly as the cube of its radius.<br />Remember: direct variation always looks like y = kx. However, we now have different variables.<br />R = k L<br />
- 22. Exercise 1:<br />Write a general equation that represents the given relationship.<br />The electric resistance, R, of a wire varies directly as its length, L.<br />The volume, V, of a sphere varies directly as the cube of its radius.<br />Remember: direct variation always looks like y = kx. However, we now have different variables.<br />R = k L<br />
- 23. Exercise 1:<br />Write a general equation that represents the given relationship.<br />The electric resistance, R, of a wire varies directly as its length, L.<br />The volume, V, of a sphere varies directly as the cube of its radius.<br />Remember: direct variation always looks like y = kx. However, we now have different variables.<br />R = k L<br />
- 24. Exercise 1:<br />Write a general equation that represents the given relationship.<br />The electric resistance, R, of a wire varies directly as its length, L.<br />The volume, V, of a sphere varies directly as the cube of its radius.<br />Remember: direct variation always looks like y = kx. However, we now have different variables.<br />R = k L<br />V = k r 3<br />
- 25. Exercise 2:<br />Write a general equation that represents the following relationship.<br />When you swim underwater the pressure, P, in your ears varies directly as the depth, d, at which you are swimming.<br />
- 26. Exercise 2:<br />Write a general equation that represents the following relationship.<br />When you swim underwater the pressure, P, in your ears varies directlyas the depth, d, at which you are swimming.<br />P = kd<br />
- 27. Exercise 2:<br />Write a general equation that represents the following relationship.<br />When you swim underwater the pressure, P, in your ears varies directlyas the depth, d, at which you are swimming.<br />P = kd<br />Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.<br />
- 28. Exercise 2:<br />Write a general equation that represents the following relationship.<br />When you swim underwater the pressure, P, in your ears varies directlyas the depth, d, at which you are swimming.<br />P = kd<br />Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.<br />P = 8.6<br />
- 29. Exercise 2:<br />Write a general equation that represents the following relationship.<br />When you swim underwater the pressure, P, in your ears varies directlyas the depth, d, at which you are swimming.<br />P = kd<br />Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.<br />P = 8.6<br />and d = 20<br />
- 30. Exercise 2:<br />Write a general equation that represents the following relationship.<br />When you swim underwater the pressure, P, in your ears varies directlyas the depth, d, at which you are swimming.<br />P = kd<br />Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.<br />P = 8.6<br />and d = 20<br />So…<br />
- 31. Exercise 2:<br />Write a general equation that represents the following relationship.<br />When you swim underwater the pressure, P, in your ears varies directlyas the depth, d, at which you are swimming.<br />P = kd<br />Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.<br />P = 8.6<br />and d = 20<br />So…<br />
- 32. Exercise 2:<br />Write a general equation that represents the following relationship.<br />When you swim underwater the pressure, P, in your ears varies directlyas the depth, d, at which you are swimming.<br />P = kd<br />Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.<br />P = 8.6<br />and d = 20<br />So…<br />
- 33. Exercise 2:<br />Write a general equation that represents the following relationship.<br />When you swim underwater the pressure, P, in your ears varies directlyas the depth, d, at which you are swimming.<br />P = kd<br />Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.<br />P = 8.6<br />and d = 20<br />So…<br />
- 34. Exercise 2:<br />Write a general equation that represents the following relationship.<br />When you swim underwater the pressure, P, in your ears varies directlyas the depth, d, at which you are swimming.<br />P = kd<br />Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.<br />P = 8.6<br />and d = 20<br />So…<br />
- 35. Exercise 2:<br />Write a general equation that represents the following relationship.<br />When you swim underwater the pressure, P, in your ears varies directlyas the depth, d, at which you are swimming.<br />P = kd<br />Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.<br />P = 8.6<br />and d = 20<br />So…<br />At an underwater depth of 80 ft, what is the pressure in your ears?<br />
- 36. Exercise 2:<br />Write a general equation that represents the following relationship.<br />When you swim underwater the pressure, P, in your ears varies directlyas the depth, d, at which you are swimming.<br />P = kd<br />Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.<br />P = 8.6<br />and d = 20<br />So…<br />At an underwater depth of 80 ft, what is the pressure in your ears?<br />We now know(from part b) that when you are underwater the constant of proportionality is k = 0.43. So our actual equation is <br />
- 37. Exercise 2:<br />Write a general equation that represents the following relationship.<br />When you swim underwater the pressure, P, in your ears varies directlyas the depth, d, at which you are swimming.<br />P = kd<br />Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.<br />P = 8.6<br />and d = 20<br />So…<br />At an underwater depth of 80 ft, what is the pressure in your ears?<br />We now know(from part b) that when you are underwater the constant of proportionality is k = 0.43. So our actual equation is <br />P = 0.43d<br />
- 38. Exercise 2:<br />Write a general equation that represents the following relationship.<br />When you swim underwater the pressure, P, in your ears varies directlyas the depth, d, at which you are swimming.<br />P = kd<br />Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.<br />P = 8.6<br />and d = 20<br />So…<br />At an underwater depth of 80 ft, what is the pressure in your ears?<br />We now know(from part b) that when you are underwater the constant of proportionality is k = 0.43. So our actual equation is <br />P = 0.43d<br />At a depth of 80 ft we would have<br />
- 39. Exercise 2:<br />Write a general equation that represents the following relationship.<br />When you swim underwater the pressure, P, in your ears varies directlyas the depth, d, at which you are swimming.<br />P = kd<br />Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.<br />P = 8.6<br />and d = 20<br />So…<br />At an underwater depth of 80 ft, what is the pressure in your ears?<br />We now know(from part b) that when you are underwater the constant of proportionality is k = 0.43. So our actual equation is <br />P = 0.43d<br />At a depth of 80 ft we would have<br />
- 40. Exercise 2:<br />Write a general equation that represents the following relationship.<br />When you swim underwater the pressure, P, in your ears varies directlyas the depth, d, at which you are swimming.<br />P = kd<br />Suppose you know that at a depth of 20 ft the pressure in your ears is 8.6 pounds per square inch. Calculate the constant of proportionality, k.<br />P = 8.6<br />and d = 20<br />So…<br />At an underwater depth of 80 ft, what is the pressure in your ears?<br />We now know(from part b) that when you are underwater the constant of proportionality is k = 0.43. So our actual equation is <br />P = 0.43d<br />At a depth of 80 ft we would have<br />At a depth of 80 ft the pressure in your ears would be 34.4 pounds per sq. inch.<br />
- 41. The three parts of exercise 2 represent a typical problem in this section. <br />Most problems follow the same basic pattern (see Blitzer textbook, page 465):<br />
- 42. The three parts of exercise 2 represent a typical problem in this section. <br />Most problems follow the same basic pattern (see Blitzer textbook, page 465):<br />Write an equation that models the statement.<br />
- 43. The three parts of exercise 2 represent a typical problem in this section. <br />Most problems follow the same basic pattern (see Blitzer textbook, page 465):<br />Write an equation that models the statement.<br />Substitute the given values in to the equation to find the value of k.<br />
- 44. The three parts of exercise 2 represent a typical problem in this section. <br />Most problems follow the same basic pattern (see Blitzer textbook, page 465):<br />Write an equation that models the statement.<br />Substitute the given values in to the equation to find the value of k.<br />Substitute the value of k into the equation from step 1.<br />
- 45. The three parts of exercise 2 represent a typical problem in this section. <br />Most problems follow the same basic pattern (see Blitzer textbook, page 465):<br />Write an equation that models the statement.<br />Substitute the given values in to the equation to find the value of k.<br />Substitute the value of k into the equation from step 1.<br />Use the equation from step 3 to answer the problem’s question.<br />
- 46. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />
- 47. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation.<br />
- 48. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation.<br />
- 49. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation.<br />
- 50. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />
- 51. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />
- 52. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />
- 53. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inchesbounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />
- 54. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />
- 55. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />
- 56. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />
- 57. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />
- 58. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />
- 59. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />
- 60. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question.<br />
- 61. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. If the ball bounced 56 in. how far was it dropped? <br />
- 62. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. If the ball bounced 56 in. how far was it dropped?<br /> In other words, if B = 56, what was d ? <br />
- 63. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. If the ball bounced 56 in. how far was it dropped?<br /> In other words, if B = 56, what was d ? <br />
- 64. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. If the ball bounced 56 in. how far was it dropped?<br /> In other words, if B = 56, what was d ? <br />
- 65. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. If the ball bounced 56 in. how far was it dropped?<br /> In other words, if B = 56, what was d ? <br />
- 66. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. If the ball bounced 56 in. how far was it dropped?<br /> In other words, if B = 56, what was d ? <br />
- 67. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. If the ball bounced 56 in. how far was it dropped?<br /> In other words, if B = 56, what was d ? <br />The ball was dropped from a height of 80 inches.<br />
- 68. Exercise 3:<br />The height that a ball bounces varies directly as the height from which it was dropped. A tennis ball dropped from 12 inches bounces 8.4 inches. From what height was the tennis ball dropped if it bounces 56 inches?<br />Step 1: Write the equation. Let B be the height the ball bounces and d the height from which the ball was dropped. <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. If the ball bounced 56 in. how far was it dropped?<br /> In other words, if B = 56, what was d ? <br />The ball was dropped from a height of 80 inches.<br />DON’T FORGET UNITS!!!<br />
- 69. Now you try!<br />
- 70. Now you try!<br />Try exercise 4 on your own. <br />
- 71. Now you try!<br />Try exercise 4 on your own. <br />Exercise 4<br />The distance required to stop a car varies directly as the square of its speed. If 200 feet are required to stop a car traveling 60 miles per hour, how many feet are required to stop a car traveling 100 miles per hour? (round your answer to the nearest foot)<br />
- 72. Now you try!<br />Try exercise 4 on your own. <br />Exercise 4<br />The distance required to stop a car varies directly as the square of its speed. If 200 feet are required to stop a car traveling 60 miles per hour, how many feet are required to stop a car traveling 100 miles per hour? (round your answer to the nearest foot)<br /> Verify that the answer is 556 feet.<br />
- 73. Inverse Variation<br /> <br />The relationship “y varies inversely as x” or “y is inverselyproportionalto x” could be written in equation form as:<br /> where k represents the constant of proportionality<br />
- 74. Inverse Variation<br /> <br />The relationship “y varies inversely as x” or “y is inverselyproportionalto x” could be written in equation form as:<br /> where k represents the constant of proportionality<br />
- 75. Inverse Variation<br /> <br />The relationship “y varies inversely as x” or “y is inverselyproportionalto x” could be written in equation form as:<br /> where k represents the constant of proportionality<br />Exercise 5:<br />Write a general equation that represents the following relationship:<br />
- 76. Inverse Variation<br /> <br />The relationship “y varies inversely as x” or “y is inverselyproportionalto x” could be written in equation form as:<br /> where k represents the constant of proportionality<br />Exercise 5:<br />Write a general equation that represents the following relationship:<br />The demand, D, for a product varies inversely as its price, P.<br />
- 77. Inverse Variation<br /> <br />The relationship “y varies inversely as x” or “y is inverselyproportionalto x” could be written in equation form as:<br /> where k represents the constant of proportionality<br />Exercise 5:<br />Write a general equation that represents the following relationship:<br />The demand, D, for a product varies inversely as its price, P.<br />
- 78. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />
- 79. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />Step 1: Write the equation.<br />
- 80. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />Step 1: Write the equation.<br />
- 81. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />
- 82. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />
- 83. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />
- 84. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />
- 85. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />
- 86. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />
- 87. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />
- 88. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />
- 89. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question.<br />
- 90. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. What is the water temperature at a depth of 5000 meters?<br />
- 91. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. What is the water temperature at a depth of 5000 meters?<br /> In other words, if d = 1000, what is T ? <br />
- 92. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. What is the water temperature at a depth of 5000 meters?<br /> In other words, if d = 1000, what is T ? <br />
- 93. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. What is the water temperature at a depth of 5000 meters?<br /> In other words, if d = 1000, what is T ? <br />
- 94. Exercise 6:<br />The water temperature, T, of the Pacific Ocean varies inversely as the water’s depth, d. At a depth of 1000 meters, the water temperature is 4.4°C. What is the water temperature at a depth of 5000 meters?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. What is the water temperature at a depth of 5000 meters?<br /> In other words, if d = 1000, what is T ? <br />The temperature is 0.88°C at a depth of 5000 m.<br />
- 95. Inverse Variation intuitively…<br />
- 96. Inverse Variation intuitively…<br />Notice what happened in the last example:<br />
- 97. Inverse Variation intuitively…<br />Notice what happened in the last example:<br />When the depth got bigger, the temperature got smaller.<br />
- 98. Inverse Variation intuitively…<br />Notice what happened in the last example:<br />When the depth got bigger, the temperature got smaller.<br />This makes sense because the farther underwater you go in the ocean the colder the temperature gets.<br />
- 99. Inverse Variation intuitively…<br />Notice what happened in the last example:<br />When the depth got bigger, the temperature got smaller.<br />This makes sense because the farther underwater you go in the ocean the colder the temperature gets.<br />In general, for inverse variation…<br />
- 100. Inverse Variation intuitively…<br />Notice what happened in the last example:<br />When the depth got bigger, the temperature got smaller.<br />This makes sense because the farther underwater you go in the ocean the colder the temperature gets.<br />In general, for inverse variation…<br /><ul><li>When one variable increases the other decreases.</li></li></ul><li>Inverse Variation intuitively…<br />Notice what happened in the last example:<br />When the depth got bigger, the temperature got smaller.<br />This makes sense because the farther underwater you go in the ocean the colder the temperature gets.<br />In general, for inverse variation…<br /><ul><li>When one variable increases the other decreases.
- 101. When one variable decreases the other increases.</li></li></ul><li>Inverse Variation intuitively…<br />Notice what happened in the last example:<br />When the depth got bigger, the temperature got smaller.<br />This makes sense because the farther underwater you go in the ocean the colder the temperature gets.<br />In general, for inverse variation…<br /><ul><li>When one variable increases the other decreases.
- 102. When one variable decreases the other increases.
- 103. Basically they do the opposite….that is why we call it inverse variation.</li></li></ul><li>Combined Variation<br />In combined variation, direct and inverse variation occur at the same time.<br />
- 104. Combined Variation<br />In combined variation, direct and inverse variation occur at the same time.<br />Example: <br />The sale of a product varies directly as its advertising budget and inversely as the price of the product.<br />
- 105. Combined Variation<br />In combined variation, direct and inverse variation occur at the same time.<br />Example: <br />The sale of a product varies directly as its advertising budget and inversely as the price of the product.<br />We could write this relationship as <br />
- 106. Combined Variation<br />In combined variation, direct and inverse variation occur at the same time.<br />Example: <br />The sale of a product varies directly as its advertising budget and inversely as the price of the product.<br />We could write this relationship as <br />Notice that since sales vary directly with advertising then if we increased our advertising budget we would expect sales to also increase.<br />
- 107. Combined Variation<br />In combined variation, direct and inverse variation occur at the same time.<br />Example: <br />The sale of a product varies directly as its advertising budget and inversely as the price of the product.<br />We could write this relationship as <br />Notice that since sales vary directly with advertising then if we increased our advertising budget we would expect sales to also increase.<br />Notice that since sales vary inversely with price then if we increased the price of the product we would expect sales to decrease.<br />
- 108. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />
- 109. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation.<br />
- 110. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people <br />
- 111. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people <br />
- 112. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people <br />Step 2: Substitute the givens to find k.<br />
- 113. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people <br />Step 2: Substitute the givens to find k.<br />
- 114. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people <br />Step 2: Substitute the givens to find k.<br />
- 115. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people <br />Step 2: Substitute the givens to find k.<br />
- 116. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people <br />Step 2: Substitute the givens to find k.<br />
- 117. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people <br />Step 2: Substitute the givens to find k.<br />
- 118. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />
- 119. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />
- 120. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question.<br />
- 121. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. How many minutes will it take 8 people to solve 24 problems?<br />
- 122. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. How many minutes will it take 8 people to solve 24 problems?<br /> In other words, if p = 8 and n = 24, what is m ? <br />
- 123. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. How many minutes will it take 8 people to solve 24 problems?<br /> In other words, if p = 8 and n = 24, what is m ? <br />
- 124. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. How many minutes will it take 8 people to solve 24 problems?<br /> In other words, if p = 8 and n = 24, what is m ? <br />
- 125. Exercise 7:<br />The number of minutes needed to finish a math assignment varies directly as the number of problems and inversely as the number of people working to solve the problems. If it takes 4 people 32 minutes to solve 16 problems, how many minutes will it take 8 people to solve 24 problems?<br />Step 1: Write the equation. Let m = number of minutes, n = number of problems, and p = number of people <br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. How many minutes will it take 8 people to solve 24 problems?<br /> In other words, if p = 8 and n = 24, what is m ? <br />It will take 24 minutes for 8 people to solve 24 problems.<br />
- 126. Joint Variation<br /> <br />The relationship “y varies jointlyas x and z” could be written in equation form as:<br /> where k represents the constant of proportionality<br />
- 127. Exercise 8:<br />The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?<br />
- 128. Exercise 8:<br />The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?<br />Step 1: Write the equation.<br />
- 129. Exercise 8:<br />The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?<br />Step 1: Write the equation.<br />
- 130. Exercise 8:<br />The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />
- 131. Exercise 8:<br />The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />
- 132. Exercise 8:<br />The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />
- 133. Exercise 8:<br />The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />
- 134. Exercise 8:<br />The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />
- 135. Exercise 8:<br />The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />
- 136. Exercise 8:<br />The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />
- 137. Exercise 8:<br />The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />
- 138. Exercise 8:<br />The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />
- 139. Exercise 8:<br />The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question.<br />
- 140. Exercise 8:<br />The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. What is the volume of a cone with B = 20 in. and h = 8 in. ? <br />
- 141. Exercise 8:<br />The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. What is the volume of a cone with B = 20 in. and h = 8 in. ? <br />
- 142. Exercise 8:<br />The Volume, V, of a cone varies jointly as the area, B, of the base and the height, h, of the cone. If V = 15.7 in.3 when B = 9.42 in. and h = 5 in., what is the volume of a cone with B = 20 in. and h = 8 in ?<br />Step 1: Write the equation.<br />Step 2: Substitute the givens to find k.<br />Step 3: Substitute k into the equation from step 1.<br />Step 4: Use the equation from step 3 to answer the question. What is the volume of a cone with B = 20 in. and h = 8 in. ? <br />The volume of a cone with B = 20 in. and h = 8 in. is approximately 53.33cubic inches.<br />

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