course Mth 163 Øô’±ßÛòĬõ¢ß±ùغ•q‡é{ðStudent Name:
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21:57:34 `q001. Note that this assignment has 5 questions If a(n) = a(n-1) + 2^n, with a(0) = 3, then substitute in turn the values 1, 2, 3 and 4 into the equation to obtain the values a(1), a(2), a(3) and a(4).
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RESPONSE --> Here is what I got for a(1)=a(1-1)+2^1 a(1)=a+2
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˜¬©ˆ´ÄQêäªføÆ”d´ÒVù‘Ÿbžq¡¥½û³æí Student Name: assignment #014
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22:09:21 `q001. Note that this assignment has 5 questions If a(n) = a(n-1) + 2^n, with a(0) = 3, then substitute in turn the values 1, 2, 3 and 4 into the equation to obtain the values a(1), a(2), a(3) and a(4).
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RESPONSE --> a(1)=a(1-1)+2^1 a(1)=a(0)+2 a(1)=a(3)+2 a(1)=5 a(2)=a(2-1)+2^2 a(2)=a(1)+4 a(2)=a(5)+4 a(2)=9 a(3)=a(3-1)+2^3 a(3)=a(2)+8 a(3)=a(9)+8 a(3)=17 a(4)=a(4-1)+2^4 a(4)=a(3)+16 a(4)=a(17)+16 a(4)=33 That's what I've got. I hope I did that right.
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22:10:28 If we substitute n = 1 into a(n) = a(n-1) + 2^n we get a(1) = a(1-1) + 2^1 or, since 1-1 = 0 and 2^1 = 2 a(1) = a(0) + 2. Since we are given a(0) = 3 we now have a(1) = 3 + 2 = 5. If we substitute n = 2 into a(n) = a(n-1) + 2^n we get a(2) = a(2-1) + 2^2 or, since 2-1 = 1 and 2^2 = 4 a(2) = a(1) + 4. Since we are given a(1) = 5 we now have a(2) = 5 + 4 = 9. If we substitute n = 3 into a(n) = a(n-1) + 2^n we get a(3) = a(3-1) + 2^3 or, since 3-1 = 2 and 2^3 = 8 a(3) = a(2) + 8. Since we are given a(2) = 9 we now have a(3) = 9 + 8 = 17. If we substitute n = 4 into a(n) = a(n-1) + 2^n we get a(4) = a(4-1) + 2^4 or, since 4-1 = 3 and 2^4 = 16 a(4) = a(3) + 16. Since we are given a(3) = 16 we now have a(4) = 17 + 16 = 33.
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RESPONSE --> Goody! I got it.
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22:21:05 `q002. If a(n) = 2 * a(n-1) + n with a(0) = 3, then what are the values of a(1), a(2), a(3) and a(4)?
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RESPONSE --> a(1)=2*a(1-1)+1 a(1)=2*a(0)+1 a(1)=2*a(3)+1 a(1)=7 a(2)=2*a(2-1)+2 a(2)=2*a(1)+2 a(2)=2*a(7)+2 a(2)=16 a(3)=2*a(3-1)+3 a(3)=2*a(2)+3 a(3)=2*a(16)+3 a(3)=35 a(4)=2*a(4-1)+4 a(4)=2*a(3)+4 a(4)=2*(35)+4 a(4)=74 Now, I hope I did this one correctly.
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22:21:39 If we substitute n = 1 into a(n) = 2 * a(n-1) + n we get a(1) = 2 * a(1-1) + 1 or since 1-1 = 0 a(1) = 2 * a(0) + 1. Since we know that a(0) = 3 we have a(1) = 2 * 3 + 1 = 7. If we substitute n = 1 into a(n) = 2 * a(n-1) + n we get a(2) = 2 * a(2-1) + 2 or since 2-1 = 1 a(2) = 2 * a(1) + 2. Since we know that a(0) = 3 we have a(2) = 2 * 7 + 2 = 16. If we substitute n = 1 into a(n) = 2 * a(n-1) + n we get a(3) = 2 * a(3-1) + 3 or since 3-1 = 2 a(3) = 2 * a(2) + 3. Since we know that a(0) = 3 we have a(3) = 2 * 16 + 3 = 35. If we substitute n = 1 into a(n) = 2 * a(n-1) + n we get a(4) = 2 * a(4-1) + 4 or since 4-1 = 3 a(4) = 2 * a(3) + 4. Since we know that a(0) = 3 we have a(4) = 2 * 35 + 4 = 74.
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RESPONSE --> O.K.!!! Yes!
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22:36:40 `q003. What are the average slopes of the graph of y = x^2 + x - 2 between the x = 1 and x= 3 points, between the x = 3 and x = 5 points, between the x = 5 and x = 7 points, and between the x = 7 and x = 9 points? What is the pattern of this sequence of slopes?
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RESPONSE --> I obtained the following coordinates by plugging the x values in the equation to obtain the y values. (1,0) (3,10) (5,28) (7,54) (9,88) The slope between x=1 and x=3 is 5. The slope between x=3 and x=5 is 9. The slope between x=5 and x=7 is 13. The slope between x=7 and x=9 is 17. The pattern is that it increases by 4 each time.
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22:38:45 At x = 1, 3, 5 , 7 and 9 we find by substituting that y = 0, 10, 28, 54 and 88. The x = 1, 3, 5, 7 and 9 points are therefore (1,0), (3,10), (5,28), (7,54) and (9,88). The run from one point to the next is always 2. The rises are respectively 10, 18, 26 and 34. The slopes are therefore slope between x = 1 and x = 3: slope = rise / run = 10 / 2 = 6. slope between x = 3 and x = 5: slope = rise / run = 18 / 2 = 9. slope between x = 5 and x = 7: slope = rise / run = 26 / 2 = 13. slope between x = 7 and x = 9: slope = rise / run = 34 / 2 = 17.
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RESPONSE --> I think that the slope between x=1 and x=3 is 5 not 6. That's a typo, I think. Other than that, it matched mine perfectly.
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22:41:46 `q004. If a solid stone sphere 4 inches in diameter weighs 3 pounds, then what would be the weight of a solid stone sphere 2 feet in diameter?
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RESPONSE --> Gosh, here we go again with these tricky questions. I think that since you are cutting the diameter exactly in half that the weight too would be exactly half. So 1.5 pounds.
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22:50:42 The volume of a sphere is proportional to the cube of its diameters, and weight is directly proportional to volume so we have the proportionality w = k d^3, where w and d stand for weight and diameter and k is the proportionality constant. Substituting the known weight and diameter we get 3 = k * 4^3, where we understand that the weight is in pounds and the diameter in inches. This gives us 3 = 64 k so that k = 3 / 64. Our proportionality equation is now w = 3/64 * d^3. So when the diameter is 2 feet, we first recall that diameter must be in inches and say that d = 24, which we then substitute to obtain w = 3/64 * 24^3. A simple calculation gives us the final weight w = 748.
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RESPONSE --> It is 748 pounds? I missed SOMETHING. Ha! We changed the feet to inches, did we change the weight somewhere too? I'm sorry, I don't understand. I understand how you got what you did from the equation, but I don't understand how one that is 4 ft and weighs 3lbs. can be proportional to one that is 2 ft. and 748 lbs, if I understood the answer correctly.
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23:30:22 `q005. Two boxes are each constructed of a single layer of cardboard. The first box is 12 inches by 18 inches by 24 inches and weighs 22 ounces; the second is 36 inches by 54 inches by 72 inches. Using proportionality determine the weight of the second box.
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RESPONSE --> I'm not sure how to begin this problem. I reviewed the linked outline and class notes 12, but can't seem to find exactly the same type of problem. I looked in our Precalculus book, pocket Super Review Precal. book, a Schuam's Precalculus Outlines book, laminated Study Guides, and now I'm out of books. Not sure where to begin. I want to find the area, but when I multiply 12x18x24, I get a l very large number.
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23:39:25 The ratios 36/12, 54/18 and 72/24 of the corresponding sides are all the same and all equal to 3, so the dimensions of the sides of the second box are 3 times those of the first. Since the thickness of the cardboard is the same on both boxes, only the dimensions of the rectangular sides change. The only thing that matters, therefore, is the surface area of the box. The proportionality is therefore of the form w = k x^2, where w is the weight of the box and x stands any linear dimension. It follows that w2 / w1 = (x2 / x1)^2. Since as we just saw x2 / x1 = 3, we see that w2 / w1 = 3^2 = 9. Since w1 = 22 oz, we write this as w2 / 22 oz = 9. Multiplying both sides by 22 oz we see that w2 = 22 oz * 9 = 198 oz.
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RESPONSE --> I worked the problem with the formula and also got 198. I only have one question, do you always divide the ratios to get the x2/x1 value? and what if they do not divide equally, I mean what if one comes out to be 5, then next 2 and so on?, or would that ever happen. If I don't need to worry about that, just tell me.
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