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# AA 2.3 & 2.9

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### AA 2.3 & 2.9

1. 1. Warm Up The length of the edge of a cube is 10 inches. How does the volume of a cube with edges 3 times as long compare to the volume of the smaller cube?
2. 2. Warm Up The length of the edge of a cube is 10 inches. How does the volume of a cube with edges 3 times as long compare to the volume of the smaller cube? The volume of the large one is 27,000 cubic inches while the smaller one is 1,000 cubic inches
3. 3. Warm Up The length of the edge of a cube is 10 inches. How does the volume of a cube with edges 3 times as long compare to the volume of the smaller cube? The volume of the large one is 27,000 cubic inches while the smaller one is 1,000 cubic inches So, the volume will be 27 times as large.
4. 4. 2.3 Fundamental Theorem of Variation & 2.9 Combined and Joint Variation
5. 5. THE ESSENTIAL QUESTION How do we solve variations? What is the Fundamental Theorem of Variation?
6. 6. The Fundamental Theorem of Variation
7. 7. The Fundamental Theorem of Variation If y varies directly as xn (y = kxn), and x is multiplied by c, then y is multiplied by cn.
8. 8. The Fundamental Theorem of Variation If y varies directly as xn (y = kxn), and x is multiplied by c, then y is multiplied by cn. If y varies inversely as xn (y = k/xn), and x is multiplied by a nonzero constant c, then y is divided by cn.
9. 9. Suppose that the value of x is doubling, how is y changing if:
10. 10. Suppose that the value of x is doubling, how is y changing if: 3 1. If y varies directly as x , then ______
11. 11. Suppose that the value of x is doubling, how is y changing if: 3 1. If y varies directly as x , then ______ y is multiplied by 8, from 23
12. 12. Suppose that the value of x is doubling, how is y changing if: 3 1. If y varies directly as x , then ______ y is multiplied by 8, from 23 2. If y varies directly as x4 , then ______
13. 13. Suppose that the value of x is doubling, how is y changing if: 3 1. If y varies directly as x , then ______ y is multiplied by 8, from 23 2. If y varies directly as x4 , then ______ y is multiplied by 16, from 24
14. 14. Suppose that the value of x is doubling, how is y changing if: 3 1. If y varies directly as x , then ______ y is multiplied by 8, from 23 2. If y varies directly as x4 , then ______ y is multiplied by 16, from 24 2 3. If y varies inversely as x , then ________
15. 15. Suppose that the value of x is doubling, how is y changing if: 3 1. If y varies directly as x , then ______ y is multiplied by 8, from 23 2. If y varies directly as x4 , then ______ y is multiplied by 16, from 24 2 3. If y varies inversely as x , then ________ y is divided by 4, from 22
16. 16. 4. The formula I = k/D2 tells that the intensity of light varies inversely as the square of the distance from the light source. What effect does doubling the distance have on the intensity of the light?
17. 17. 4. The formula I = k/D2 tells that the intensity of light varies inversely as the square of the distance from the light source. What effect does doubling the distance have on the intensity of the light? We have an inverse variation so our answer is divided by cn .
18. 18. 4. The formula I = k/D2 tells that the intensity of light varies inversely as the square of the distance from the light source. What effect does doubling the distance have on the intensity of the light? We have an inverse variation so our answer is divided by cn . In the formula D is squared, so our n value is 2, since we are doubling, our c value is 2.
19. 19. 4. The formula I = k/D2 tells that the intensity of light varies inversely as the square of the distance from the light source. What effect does doubling the distance have on the intensity of the light? We have an inverse variation so our answer is divided by cn . In the formula D is squared, so our n value is 2, since we are doubling, our c value is 2. So we would divide by 4, which is the same as multiplying by 1/4, so the light is 1/4 the intensity.
20. 20. Combined & Joint Variation
21. 21. Combined & Joint Variation Combined variation - when direct and inverse variations occur together
22. 22. Combined & Joint Variation Combined variation - when direct and inverse variations occur together
23. 23. Combined & Joint Variation Combined variation - when direct and inverse variations occur together Joint variation - when one quantity varies directly as the product of two or more independent variables
24. 24. Combined & Joint Variation Combined variation - when direct and inverse variations occur together Joint variation - when one quantity varies directly as the product of two or more independent variables
25. 25. Combined & Joint Variation Combined variation - when direct and inverse variations occur together Joint variation - when one quantity varies directly as the product of two or more independent variables
26. 26. Combined & Joint Variation Combined variation - when direct and inverse variations occur together kx for example - y= z Joint variation - when one quantity varies directly as the product of two or more € independent variables
27. 27. Combined & Joint Variation Combined variation - when direct and inverse variations occur together kx for example - y= z Joint variation - when one quantity varies directly as the product of two or more € independent variables for example - A = kbh
28. 28. 5. A baseball pitcher’s earned run average (ERA) varies directly as the number of earned runs allowed and inversely as the number of innings pitched. Write a general equation to model this situation.
29. 29. 5. A baseball pitcher’s earned run average (ERA) varies directly as the number of earned runs allowed and inversely as the number of innings pitched. Write a general equation to model this situation. Let e = ERA, R = # of earned runs and I = # of innings.
30. 30. 5. A baseball pitcher’s earned run average (ERA) varies directly as the number of earned runs allowed and inversely as the number of innings pitched. Write a general equation to model this situation. Let e = ERA, R = # of earned runs and I = # of innings. kR E= I €
31. 31. 5. A baseball pitcher’s earned run average (ERA) varies directly as the number of earned runs allowed and inversely as the number of innings pitched. Write a general equation to model this situation. Let e = ERA, R = # of earned runs and I = # of innings. kR E= I 6. In a recent year, a pitcher had an ERA of 2.56, having given up 72 earned runs in 253 innings. How many earned runs would the pitcher have given up if he had pitched 300 innings, assuming € that his ERA remained the same?
32. 32. 5. A baseball pitcher’s earned run average (ERA) varies directly as the number of earned runs allowed and inversely as the number of innings pitched. Write a general equation to model this situation. Let e = ERA, R = # of earned runs and I = # of innings. kR E= I 6. In a recent year, a pitcher had an ERA of 2.56, having given up 72 earned runs in 253 innings. How many earned runs would the pitcher have given up if he had pitched 300 innings, assuming € that his ERA remained the same? kR E= I €
33. 33. 5. A baseball pitcher’s earned run average (ERA) varies directly as the number of earned runs allowed and inversely as the number of innings pitched. Write a general equation to model this situation. Let e = ERA, R = # of earned runs and I = # of innings. kR E= I 6. In a recent year, a pitcher had an ERA of 2.56, having given up 72 earned runs in 253 innings. How many earned runs would the pitcher have given up if he had pitched 300 innings, assuming € that his ERA remained the same? kR 72k E= 2.56 = I 253 € €
34. 34. 5. A baseball pitcher’s earned run average (ERA) varies directly as the number of earned runs allowed and inversely as the number of innings pitched. Write a general equation to model this situation. Let e = ERA, R = # of earned runs and I = # of innings. kR E= I 6. In a recent year, a pitcher had an ERA of 2.56, having given up 72 earned runs in 253 innings. How many earned runs would the pitcher have given up if he had pitched 300 innings, assuming € that his ERA remained the same? kR 72k 72k E= 2.56 = 253⋅ 2.56 = ⋅ 253 I 253 253 € € €
35. 35. 5. A baseball pitcher’s earned run average (ERA) varies directly as the number of earned runs allowed and inversely as the number of innings pitched. Write a general equation to model this situation. Let e = ERA, R = # of earned runs and I = # of innings. kR E= I 6. In a recent year, a pitcher had an ERA of 2.56, having given up 72 earned runs in 253 innings. How many earned runs would the pitcher have given up if he had pitched 300 innings, assuming € that his ERA remained the same? kR 72k 72k E= 2.56 = 253⋅ 2.56 = ⋅ 253 647.68 = 72k I 253 253 € € € €
36. 36. 5. A baseball pitcher’s earned run average (ERA) varies directly as the number of earned runs allowed and inversely as the number of innings pitched. Write a general equation to model this situation. Let e = ERA, R = # of earned runs and I = # of innings. kR E= I 6. In a recent year, a pitcher had an ERA of 2.56, having given up 72 earned runs in 253 innings. How many earned runs would the pitcher have given up if he had pitched 300 innings, assuming € that his ERA remained the same? kR 72k 72k E= 2.56 = 253⋅ 2.56 = ⋅ 253 647.68 = 72k k ≈9 I 253 253 € € € € €
37. 37. 5. A baseball pitcher’s earned run average (ERA) varies directly as the number of earned runs allowed and inversely as the number of innings pitched. Write a general equation to model this situation. Let e = ERA, R = # of earned runs and I = # of innings. kR E= I 6. In a recent year, a pitcher had an ERA of 2.56, having given up 72 earned runs in 253 innings. How many earned runs would the pitcher have given up if he had pitched 300 innings, assuming € that his ERA remained the same? kR 72k 72k E= 2.56 = 253⋅ 2.56 = ⋅ 253 647.68 = 72k k ≈9 I 253 253 9R 2.56 = 300 € € € € €
38. 38. 5. A baseball pitcher’s earned run average (ERA) varies directly as the number of earned runs allowed and inversely as the number of innings pitched. Write a general equation to model this situation. Let e = ERA, R = # of earned runs and I = # of innings. kR E= I 6. In a recent year, a pitcher had an ERA of 2.56, having given up 72 earned runs in 253 innings. How many earned runs would the pitcher have given up if he had pitched 300 innings, assuming € that his ERA remained the same? kR 72k 72k E= 2.56 = 253⋅ 2.56 = ⋅ 253 647.68 = 72k k ≈9 I 253 253 9R 9R 2.56 = 300 ⋅ 2.56 = ⋅ 300 300 € 300 € € € €
39. 39. 5. A baseball pitcher’s earned run average (ERA) varies directly as the number of earned runs allowed and inversely as the number of innings pitched. Write a general equation to model this situation. Let e = ERA, R = # of earned runs and I = # of innings. kR E= I 6. In a recent year, a pitcher had an ERA of 2.56, having given up 72 earned runs in 253 innings. How many earned runs would the pitcher have given up if he had pitched 300 innings, assuming € that his ERA remained the same? kR 72k 72k E= 2.56 = 253⋅ 2.56 = ⋅ 253 647.68 = 72k k ≈9 I 253 253 9R 9R 2.56 = 300 300 ⋅ 2.56 = ⋅ 300 300 € R ≈ 85 runs € € € €
40. 40. 7. The power in an electrical circuit varies jointly as the square of the current and the resistance. Write a formula to show this relationship.
41. 41. 7. The power in an electrical circuit varies jointly as the square of the current and the resistance. Write a formula to show this relationship. Let p = power, c = current and r = resistance.
42. 42. 7. The power in an electrical circuit varies jointly as the square of the current and the resistance. Write a formula to show this relationship. Let p = power, c = current and r = resistance. Then P = kc2r
43. 43. 7. The power in an electrical circuit varies jointly as the square of the current and the resistance. Write a formula to show this relationship. Let p = power, c = current and r = resistance. Then P = kc2r b. The power in a certain circuit is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power in that circuit when the current is 20 amps and the resistance is 21 ohms.
44. 44. 7. The power in an electrical circuit varies jointly as the square of the current and the resistance. Write a formula to show this relationship. Let p = power, c = current and r = resistance. Then P = kc2r b. The power in a certain circuit is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power in that circuit when the current is 20 amps and the resistance is 21 ohms. HINT: Find k ﬁrst.
45. 45. 7. The power in an electrical circuit varies jointly as the square of the current and the resistance. Write a formula to show this relationship. Let p = power, c = current and r = resistance. Then P = kc2r b. The power in a certain circuit is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power in that circuit when the current is 20 amps and the resistance is 21 ohms. HINT: Find k ﬁrst. so... 1500 = k(15)2(10)
46. 46. 7. The power in an electrical circuit varies jointly as the square of the current and the resistance. Write a formula to show this relationship. Let p = power, c = current and r = resistance. Then P = kc2r b. The power in a certain circuit is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power in that circuit when the current is 20 amps and the resistance is 21 ohms. HINT: Find k ﬁrst. so... 1500 = k(15)2(10) 1500 = k(2250)
47. 47. 7. The power in an electrical circuit varies jointly as the square of the current and the resistance. Write a formula to show this relationship. Let p = power, c = current and r = resistance. Then P = kc2r b. The power in a certain circuit is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power in that circuit when the current is 20 amps and the resistance is 21 ohms. HINT: Find k ﬁrst. so... 1500 = k(15)2(10) 1500 = k(2250) 2/3 = k
48. 48. 7. The power in an electrical circuit varies jointly as the square of the current and the resistance. Write a formula to show this relationship. Let p = power, c = current and r = resistance. Then P = kc2r b. The power in a certain circuit is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power in that circuit when the current is 20 amps and the resistance is 21 ohms. HINT: Find k ﬁrst. NOW: Find the power. so... 1500 = k(15)2(10) 1500 = k(2250) 2/3 = k
49. 49. 7. The power in an electrical circuit varies jointly as the square of the current and the resistance. Write a formula to show this relationship. Let p = power, c = current and r = resistance. Then P = kc2r b. The power in a certain circuit is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power in that circuit when the current is 20 amps and the resistance is 21 ohms. HINT: Find k ﬁrst. NOW: Find the power. so... 1500 = k(15)2(10) P = (2/3)c2r 1500 = k(2250) 2/3 = k
50. 50. 7. The power in an electrical circuit varies jointly as the square of the current and the resistance. Write a formula to show this relationship. Let p = power, c = current and r = resistance. Then P = kc2r b. The power in a certain circuit is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power in that circuit when the current is 20 amps and the resistance is 21 ohms. HINT: Find k ﬁrst. NOW: Find the power. so... 1500 = k(15)2(10) P = (2/3)c2r 1500 = k(2250) P = (2/3)(20)2(21) 2/3 = k
51. 51. 7. The power in an electrical circuit varies jointly as the square of the current and the resistance. Write a formula to show this relationship. Let p = power, c = current and r = resistance. Then P = kc2r b. The power in a certain circuit is 1500 watts when the current is 15 amps and the resistance is 10 ohms. Find the power in that circuit when the current is 20 amps and the resistance is 21 ohms. HINT: Find k ﬁrst. NOW: Find the power. so... 1500 = k(15)2(10) P = (2/3)c2r 1500 = k(2250) P = (2/3)(20)2(21) 2/3 = k P = 5600 watts
52. 52. Homework worksheet 2.3A/2.9A